Does suppressing a desire make it more powerful? Based on my minimal knowledge of old Freudian Psychoanalysis, if our drives and urges are not met they become more powerful, even overwhelming, for the individual, perhaps causing them psychological difficulties or even causing them to then project the restriction on others. This sentiment has been echoed many times in different contexts since then and it seems to me that the modern generation operates under the assumption that trying too hard to suppress a desire is unhealthy (sexual liberation, free expression etc). It is often used, for example, as a criticism of religion that attempts to impose rules and boundaries which result in the suppression of urges and even further curiosity simply because it is forbidden. I also see many pop psychology articles about this idea. 

My question is: what empirical evidence exists to support the notion that repressed desires will make them stronger? Other relevant question that may be of interest: Is it a result of human's natural curiosity? Is it intrinsic to human drive? Is it a form of obsessive-compulsive behavior? What criticism has been leveled against this theory in the Freudian version or any modern interpretations?
First of all, your question title asks about suppression and your question in the main text asks


  what empirical evidence exists to support the notion that repressed desires will make them stronger?


What you need to be careful about here is that repression is different to suppression.

Repression is unconscious/subconscious, whereas suppression is completely conscious


  repression is the term for unintentional, unconscious forgetting,
  and suppression [is] intentional, conscious removal of a thought from subsequent conscious attention. Suppression is almost never studied, whereas a wide literature has formed around the concept of repression.....
  
  ....Although repression and suppression processes are by no means mutually exclusive or contradictory, the interest value of repression has tended to exclude suppression from scientific study. It is in this sense that Freud was responsible for what we do not know about suppression. (Wegner, 1992)


Short Answer

Wegner, in the above referenced journal article points out a lot of evidence to back the though that any attempt to suppress thoughts or desires will be in vain as they will rebound.  Whether or not they become stronger is open to interpretation, as it is concluded that


  When the Rolling Stones sang “You can’t always get what you want,” they provided a bit of relief by adding “but if you try some time, you just might find, you get what you need.” Regrettably, their refrain rings a false note in the world of thought suppression. The research to date suggests that you can’t always think what you want, and that even if you need ever so dearly to stop thinking of something, you still may not get what you need.


Long Answer

As I said in my answer to a previous question,


  With Sigmund Freud, lot of people get hung up on the sexual elements of his theories when really, Freud’s theories centres on his motivation and drive theory and they centre on all sorts of pleasure. We are motivated towards whatever brings pleasure and are motivated away from pain, difficulty, destruction, confusion, etc.


When looking for empirical evidence, Freud's work itself is derived from empirical evidence.


  His theories are clinically derived - i.e. based on what his patients told him during therapy. (McLeod, 2007)


Repression is described by Freud by saying


  the essence of repression lies simply in turning something away, and keeping it at a distance, from the conscious. This view of repression would be made more complete by assuming that, before the mental organization reaches this stage, the task of fending off instinctual impulses is dealt with by the other vicissitudes which instincts may undergo - e.g. reversal into the opposite or turning round upon the subject’s own self. (Freud, 1915/1957)


There are 2 parts to repression,

Primal Repression


  a first phase of repression, which consists in the psychical (ideational) representative of the instinct being denied entrance into the conscious. With this fixation is established; the representative in question persists unaltered from then onwards and the instinct remains attached to it. (Freud, 1915/1957)


Repression Proper


  second stage of repression, repression proper, affects mental derivatives of the repressed representative, or such trains of thought as, originating elsewhere, have come into associative connection with it. On account of this association, these ideas experience the same fate as what was primally repressed. Repression proper, therefore, is actually an after-pressure [Nachdrängen]. (Freud, 1915/1957)


You Can't Always Think What You Want: Problems in the Suppression of Unwanted Thoughts (Wegner, 1992) is an article covering articles etc. from Freud onwards.

What you need to be careful about here is that repression is different to suppression.

Repression is unconscious/subconscious, whereas suppression is completely conscious


  repression is the term for unintentional, unconscious forgetting,
  and suppression [is] intentional, conscious removal of a thought from subsequent conscious attention. Suppression is almost never studied, whereas a wide literature has formed around the concept of repression.....
  
  ....Although repression and suppression processes are by no means mutually exclusive or contradictory, the interest value of repression has tended to exclude suppression from scientific study. It is in this sense that Freud was responsible for what we do not know about suppression. (Wegner, 1992)


Now think about suppression in a quote provided by Wegner in the above article.


  Try to pose for yourself this task: not to think of a polar bear, and you will see that the cursed thing will come to mind every minute. (Fyodor Dostoevsky, Winter Notes on Summer Impressions)


Conscious efforts to suppress unwanted thoughts are reported by normal individuals (Rachman &amp; de Silva, 1978), as well as by those suffering from problems such as


depression (Turner, Beidel, &amp; Nathan, 1985),
anxiety and worry (Borkovec, Robinson, Pruzinsky, &amp; DePree, 1983),
obsessions and compulsions (Rachman &amp; Hodgson, 1980; Reed, 1985),
addiction (Marlatt &amp; Parks, 1982),
obesity (Polivy, 1990), and
posttraumatic stress (Ellis, 1983; Horowitz, 1975; Pennebaker, 1988; Tait &amp; Silver, 1989)


and is employed for the purposes of action inhibition (behavioural self-control), communication inhibition (e.g. keeping secrets), and emotional inhibition (e.g. to remain calm in stressful situations) (Wegner, 1992).

The suppression cycle

The exercise of continuous thought suppression is reminiscent of the task of
Sisyphus, the figure of Greek myth who was condemned forever to push a stone up a mountain only to have it roll back down. As a rule, people asked to suppress a thought (e.g., of a white bear) while they deliver verbal reports of their thinking show a cyclic sequence of activities with precisely this up-and-down character (Wegner et al., 1987). Suppression begins with a plan to self-distract (e.g., “Okay, I’ll think of something else”), continues with the choice of a distracter (e.g., “I’ll think about the telephone”) and a period of concentration on distracter-relevant thoughts (e.g., “This phone is a portable, etc.”), and is completed with an intrusive return of the suppressed thought (e.g., “There’s the white bear again”). The cycle then repeats with a return to the plan to self-distract (i.e., “Okay, I’ll think of something else”).

Interestingly, according to Wegner (1992), Knutson (1990) found, that people asked to shut their eyes and report their thinking during suppression indicate more frequent occurrences of the unwanted thought than those allowed to keep their eyes open.

Automatic Target Search

This is the operation of an automatic search for the unwanted thought during suppression.  For example, 


  an undergraduate student I asked to think aloud while she tried not to think of the Statue of Liberty began talking about the books on the wall of the room and in a matter of moments mentioned that one was just the greenish tint of the Statue (Wegner, 1992).


All of the above information points to the fact that any attempt to suppress thoughts or desires will be in vain as they will rebound.


  Suppression is a street battle raging in the forefront of the mind, not a few random muggings hidden deep in its alleyways (Wegner, 1992).


References

Borkovec, T. D., Robinson, E., Pruzinsky, T., &amp; DePree, J. A. (1983). Preliminary exploration of worry: Some characteristics and processes. Behaviour Research and Therapy, 21: pp. 9—16.PMID: 6830571 

Ellis, E. M. (1983). A review of empirical rape research: Victim reactions and response to treatment. Clinical Psychology Review. 3: pp. 473—490.DOI: 10.1016/0272-7358(83)90025-9

Freud, S. (1957). Repression. In: J. Stachey (Ed.), The standard edition of the complete psychological works of Sigmund Freud (Vol. 14, pp. 146—158) London: Hogarth. (Original work published 1915)

Horowitz, M. (1975). Intrusive and repetitive thoughts after experimental stress. Archives of General Psychiatry, 32: pp. 1457—1463.DOI: 10.1001/archpsyc.1975.01760290125015

Knutson, B. (1990). Thought suppression and access to distraction. Unpublished research data, Stanford University, Stanford, CA.Available by emailing: knutson'at'stanford.edu

Marlatt, G. P., &amp; Parks, G. A. (1982). Self-management of addictive behaviors. In: P. Karoly &amp; F. H. Kanfer (Eds.), Self-management and behavior change (pp. 443—488). New York: Pergamon.

McLeod, S. (2007). Psychodynamic Approach. Simply Psychology [Online]Available at: https://www.simplypsychology.org/psychodynamic.html

Pennebaker, J. W. (1988). Confession, inhibition, and disease. In: L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 22). San Diego: Academic Press.

Polivy, J. (1990). Inhibition of internally-cued behavior. In: E. T. Higgins &amp; R. M. Sorrentino (Eds.), Handbook of motivation and cognition (Vol. 2, pp. 131—147). New York: Guilford.

Rachman, S., &amp; de Silva, P. (1978). Abnormal and normal obessions. Behaviour Research and Therapy, 16: pp. 233—248DOI: 10.1016/0005-7967(78)90022-0

Rachman, S., &amp; Hodgson, R. J. (1980). Obsessions and compulsions. Englewood Cliffs, NJ: Prentice-Hall.

Reed, G. F. (1985). Obsessional experience and compulsive behavior. Orlando, FL: Academic Press.

Tait, R., &amp; Silver, R. C. (1989). Coming to terms with major negative life events. In J. S. Uleman &amp; J. A. Bargh (Eds.), Unintended thought (pp. 351—382). New York: Guilford.

Turner, S . M., Beidel, D. C., &amp; Nathan, R. S . (1985). Biological factors in obsessive-compulsive disorders. Psychological Bulletin, 97: pp. 430—450.DOI: 10.1037/0033-2909.97.3.430

Wegner, D. M., Schneider, D. J., Carter, S., 111, &amp; White, L. (1987). Paradoxical effects of thought suppression. Journal of Personality and Social Psychology, 53: pp. 5—13.DOI: 10.1037/0022-3514.53.1.5

Wegner, D. M. (1992). You Can't Always Think What You Want: Problems in the Suppression of Unwanted Thoughts. Advances in Experimental Social Psychology, 25: pp. 193—225DOI: 10.1016/S0065-2601(08)60284-1