What will convince my mom to stop treating me like a baby?

[JustDance7]

Just try your best to show her how mature you are. Firstly, don't complain out loud about these matters, instead accept the fact that your mother knows best. This alone might show her that you are old enough for these things. Maybe ask her if you can try out a tankini or monokini, which or two types of swim suits that are a little more revealing than a one-piece, but still cover a large amount of skin. As for the Facebook accounts, try suggesting that you two share an account until she trusts you enough to have your own. Even then, allow her to have frequent looks over your wall, friend list, etc. For the bra, simply tell her that you are developing into a woman and need one. If she refuses to let you get one, tell a trusted female family member, and maybe they can talk some sense into her.
Good luck!