How do I install switches in this box?
Sounds like you have one black hot wire coming into the box (probably that bottom one), which passed through a hot over to your bedroom switch.

So my guess would be that you should have two of the blacks wirenutted together -- that would be the hot wire coming in to the box, and the passthrough to the bedroom light. Also in that bundle should be two pigtailed wires that go to one pole of your 2 switches.

Then the other pole of the switches would be connected to the remaining black wires. The white neutrals should remain bundled together as they are. 

It looks like the ground wires also have loops in them for connecting to screw terminals, you should make sure that all of the ground wires in the box are connected together with a wire nut so you have an uninterrupted ground even if there are no switches connected, then pigtail off of that bundle to connect to the ground screw on your new switches.

Here's a diagram showing my guess at how the hots should be connected:



Though this is only a guess -- an easy way to identify which one is the hot wire would be to tape or wirenut the exposed ends of the wire and leave them hanging out of the box, turn on the breaker, then use your non-contact voltage sensor to see which one is hot.

There's no easy way other than trial-and-error to identify which ones go to each bathroom light and which one goes over to the bedroom.

Oh, and do remember to connect the ground to the new switches since you have a plastic box (even if the previous switches were not grounded) - I think it's required by code now, but even if not, it's a good idea -  especially in a bathroom where wet hands will be touching the light switch.