How to use 弄 in a sentence?
My impression as a learner is that 弄 functions as a unspecified verb.  Basically, we take a sentence and replace a specific verb (or multiple verbs) by the unspecific 弄.

弄 is typically used within a result complement such as 弄到, 弄坏, 弄丢, 弄清楚, and so on.  I'll focus on this common usage.  In this way 弄 offers a short-cut to the result without specifying the verb which caused it.  (It's also used in 弄得怎么样 and 怎么弄.)

In English, an unspecified verb would normally be phrased "to do", which is sometimes an appropriate translation.  We'd use it in English in the phrase "get my hair done" in place of "get my hair cut", "get my hair dyed", etc.

Why do this?  This is probably an incomplete list but here's some reasons...

1. Ignorance of which verb is correct

Let's start with this example:


  客人打坏电视机[src]
  The guest broke [by bashing] the TV


In Chinese, the verb is 打 = "hit" and the result is 坏 = "break".

Now suppose we didn't know how the guest broke the TV, only that they broke the TV.  Maybe they spilled a drink on the TV.  Maybe they installed some software on it that broke it.  Since we don't know what verb to use, we might use the unspecific 弄:


  version 1: 客人弄坏了电视机
  version 2: 客人把电视机弄坏了[src]
  The guest broke [in some unknown way] the TV


Note: It's common to use 把 in conjunction with 弄 as above (see: Using "ba" sentences).

Other examples:


  男子弄丢共享单车被索赔2.55万[src]
  Boy lost [in some unknown way] a public bike; asked for 25,500 yuan in compensation
  
  小孩不慎弄瞎他人眼睛[src]
  Kid inattentively blinded [in some unknown way] another person's eyes


2. Multiple ways of doing something

This is similar to how "done" is used in "get your hair done" in English.  If you go to a hairdresser, they don't immediately know what verb is appropriate for your hair.  Maybe "dye" or "cut" or some other verb applies.  Maybe multiple verbs are needed (e.g. the customer wants their hair both cut and dyed).  For example:


  怎么弄头发好看又简单图片[src]
  How to do hair, both good looking and simple: photos
  
  破案了!韦世豪为何要弄这样的新发型?[src]
  Solved! Why Wei Shihao wants to do this kind of new hairstyle.


Other examples are:


  怎么才能弄清楚自己想要什么?[src]
  How to be able to make clear [in some way] what you want?
  
  离职时,一定要弄明白这四点,不然只会越跳槽越糟糕[src]
  When leaving office, [you] must understand [in some way] these four points, or else it will result in more job-hopping and more bad luck


We "make something clear" or "understand" = "make something understood (to yourself)", but there's not a singly way of achieving this.

3. To encompass all possibly ways of doing something

Let's start here:


  别把衣服洗坏了
  Don't damage your clothes by washing them.
  
  别把衣服穿坏了
  Don't damage your clothes by wearing them out.
  
  别把衣服剪坏了
  Don't damage your clothes by cutting them.


We can say all of these (and more) at once using:


  别把衣服弄坏了
  Don't damage [in any way] your clothes.


Other examples:


  在这个球馆打我的最后一场比赛，你们别把我弄哭，因为我现在很高兴。[src]
  In this stadium, I played my last match, don't make me cry [in any way], because I am now happy


4. To withhold this information


  沙溢把郭敬明弄哭了！[src]
  Sha Yi made [in some unspecified way] Guo Jingmeng cry


You need to read the article to understand how Sha Yi made Guo Jingmeng cry.


  你已经弄乱我的心，什么时候弄乱我的床[src]
  You've already messed up [in some unspecified way] my heart, when will you mess up [in some unspecified way] my bed?


Cheeky!



So let's try it on the given examples:



  Don't make the child cry.
  别把孩子弄哭了


I found this used here.

  Loosen the belt.
  把腰带弄宽松一些


The closest I found online is:


  新买的t恤袖子有点紧怎么弄宽松一点[src]
  Newly-bought T-shirt sleeves are a little tight, how to loosen them?


  There is one more problem that has yet to be clarified.
  还有一个问题没有弄清楚


I found this used here.