Location of diode bridge on transformer circuit?
You're confusing the "oldfashioned" linear mains supply like this:



Reasons to use this design are:


It is a simple design
As long as you don't touch the primary side of the transformer, it's pretty safe


With the more modern switched mode power supply: 



As you can see the switched mode supply is much more complex !

Reasons to use this design are:


the transformer is used a a much higher frequency than the 50 or 60
Hz mains frequency meaning that it can be much smaller and more
efficient
the smoothing capacitors after the transformer can be smaller
being a switched supply means that it is much more efficient
it can be made more compact (because of reason one)
it can be made cheaper (no expensive transformer)
lighter in weight, easier to carry.


For Bonus points:

Here's a "circuit level" example of a simple switched mode mains supply, note that it has 2 outputs, 5 V and 12 V. You could leave out the 12 V output to make it even more simple.



Now compare that to the first schematic !