Sound deadening drywall
That's key. If you hang sound deadening drywall from metal studs staggered between the existing wood studs it helps, since there is less mechanical transfer than wood studs.Use staggered 2x4 studs on a 2x6 base plate if you can fit it. Pack the cavity with fiberglass batts. Insulation helps, sound deadening drywall helps, reducing the acoustical bridging is a BIG help. If you aren't messing with the other side of the bathroom walls then you could just tack a shim piece to the inside and add new studs on the inside only.
I'd agree with just insulation doing little to nothing...the resilient channel is what separates the two layers of drywall from being a vibration microphone for each other (think of the kid's styrofoam cup & string phone...yarn transmits sound great...). The resilient channel separates the two surfaces, creating a sound barrier. Does it soundproof the room? absolutely not...but it sure does help!I remodeled my own bathroom last winter and insulated all the walls....I changed my floor from vinyl to tile and put in a solid core door....its probably louder than my old bath..its like a eco chamber in there....I think all the sound goes through the door..doesn't matter how much you insulate
What?#2 post is the answer, Also cast iron pipes. Fiber glass batts have no real sound deadening efffect