How can I wire 3 Lutron Caseta dimmer switches in bathroom?
We gotta do something about that mess

There's no way to avoid the fact that this many wires will be a huge mess in this box.  It's already a huge mess and efforts were made to keep it neat.    

Your #1 problem is just what you're running into - wires are not well-marked.  You've done yourself a huge favor by identifying everything before you tore it all apart.  But now let's mark every single wire before we go any further.  Get some colored tape.  

Color coding

Wires themselves are color-coded thusly. 


Green for ground (mandatory)
White or gray for neutral (mandatory).  
Hot - any other color.  It's convenient to use black for always-hot, or a second color if you have 2 circuits in there, which mercifully you do not. 


Wires can also be marked with black or colored tape.  If a wire is a hot color, marking it with tape is only an identifier.   If in cable, marking a neutral with a hot color makes it a hot.  This probably won't come up in  your case, just don't do that.   Putting white, gray or green on a hot colored wire does NOT make it a neutral or ground (unless the wire is quite large, like #00 or larger).  So feel free to use white or green to mark a hot. 

Anytime you mark a wire, take the time to mark it on both ends - otherwise you'll lose your mind later. 

Mark those hots

This is what's really happening.  Notice it's fairly straightforward when you untangle the wires. 



Right now, you have supply coming in, and 4 loads going out.  Supply neutral is simply nutted with the 4 load neutrals in a big 5-bundle.  Grounds all go to those plentiful and convenient ground screws in the back of the box - you'll be adding more to those, as many of your switches will want ground and some will NEED ground.  The screws that hold it down, are not good enough as a ground path. 

Then your supply "hot" goes to every single switch.  It looks like in some cases, like #3, it uses both a screw and a backstab to make the connections. The screw-backstab are connected together internally.  If you have a non-contact voltage tester (or a contact one), this will be the terminal that is always hot, even when all the switches are off.   This is the same wire.  I recommend you color-code this one black.    Which it already is in most cases.  Mark it with black tape otherwise.  Be very clear which cable this comes from, I would even go ahead and mark the black wire with black tape to make it clear. 

Then, off each switch, is a "switched-hot" wire going to the load.  This is the one you want to carefully mark.   This wire is only energized when the switch for that load is on.  I want you to use your colored electrical tape to mark every single one of them something other than black.  When you get pigtails on the switches, mark them too.   Now it looks like this 



Every single wire has a color, and the color matches its purpose. 

This is the key to not losing your mind in a box like this. 

Adding those smart switches

It's going to get more messy with the smart switches.  Some of them will need neutral.   That neutral bundle already has 5 wires in it, we'll ned to add one more - a short pigtail, maybe 6" or so.  That pigtail will go to another new wirenut which will take all the neutrals from the smart switches.  You can get special connectors designed to take as many as 8 wires, if you can't get 6 in a wire nut.

The smart switches may also dangle unused pigtails, but you're familiar with that. 

End up looking like this