I wouldn't tear the bathroom wall apart... I'd tear the bedroom wall apart. What you are experiencing is the noise traveling through mechanical connections to the wall (screwed to studs) and the noise from the water passing through thin copper pipe (probably the Type M pipe rather than the thicker Type L).
The only effective way I know of is to insulate that wall with Roxul Safe and Sound. It's an insulation designed for noise.... and noise only. It doesn't burn. To go the added mile, use resilient channels before hanging the drywall. The demo is probably a DIY, the installation of the Roxul is DIY, and the drywall.... call someone. For a few hundred bucks the repair will disappear.
Edit:
Blown in insulation will not work... blown in is for warmth, not sound. It's the sound transfer you're trying to conquer. We once built a house and insulated the walls with thermal insulation, thinking just like you are. It didn't work. We had the type L pipe, all the plumbing was fastened with cushioned fasteners. We thought we had it knocked. Turned out, it was a waste of money.
If you truly want to fix it, just take down the drywall and do it the right way with the Roxul. Going into a wet wall from the bathroom side is a suicide mission. Building an offset wall will create a door opening that's 10"... with a door that has a 5" jamb. Then there's the carpet... or other flooring surface. It sounds like a difficult job, but it's probably only one weekend.
Otherwise, just buy your son some foam ear plugs at Costco and call it a day.
Sounds like you pulled out some butterfly anchors instead of unscrewing them, or even large molly type.
What I do in situations like this is to install a decorative BOARD behind the towel rack or toilet paper holder, on the surface of the drywall, which screws directly into studs. This can simply be a piece of 1X4 with routed or finished edges, or baseboard or crown molding in 3 inch or larger width, painted white or the wall color. If desired, continue around the room at a single height, making this look like a standard feature similar to the top of chair rail decoration.
Install this with screws long enough to reach into the studs behind the drywall, I typically use 3 inch long self drilling screws. If you don't want the screw heads to show, spackle over them or use caulk before painting. The baseboard or molding typically can be found in 3 inch or larger widths, cut to length as needed, providing enough room to install the racks or fixtures on it. The racks can then can screw into this. This eliminates any unsightly holes or bad patching that might be present, as the board goes right over them. Studs are typically 16 inches on center, use a stud finder to locate them. For the toilet paper holder, if the board is attached by screws into one stud firmly, and the other end is installed with butterfly anchors, it should never come off the wall.
The exception would be a block wall behind the drywall, which probably would have furring strips holding it up. You'd then be able to use a drill and concrete anchors with longer screws to hold up either the board, or the racks themselves.
Best Answer
Do you mean purple rather than red drywall? I see that you are from Spain, so I don't know if this answer will apply in your country - but here in the USA, the difference is that green drywall is moisture-resistant, but it is not waterproof. Purple drywall has superior moisture- and mold-resistant characteristics that make it much more suitable for use where it might be in contact with water.