Why does my shower turn on when I turn on the hot water in the bathroom sink?
Sounds like a washerless faucet issue.

Water pressure holds the plate that seals the unit. As the pressure drops, the handle can move, allowing flow. You may have luck replacing the seals--not very difficult. Turn the water both hot and cold off. With most of these type faucets the top section unscrews. Remove the cartridge take the seals to a home store or hardware store and they will usually have replacements (there are different sizes). Reassemble and see if that fixes the problem. 

Other than that, a lighter weight handle may be needed if you cannot tighten it up.