That is seriously embarrassing! Why would your mum do that? Maybe you could take a close (girl) friend or maybe a cousin who you are close to. You should be able to find some pretty cheap bras. If you don't have a friend who you feel comfortable with, buy a few vest-tops. You can get really thin ones, so they wont feel weird if you wear them under your normal clothes. That should stop the see-through problem.
I dont know how old you are, but when you get to about 16 or 17 you'll probably have a few really close gal-pals who wont mind going with you. Honestly, though, you could go by yourself, if you feel comfortable. If you can survive these next few years, you'll be off to college and you wont need your mum to go with you. And hopefully by that point your mum will realise how grown up you are and will stop stop acting like this.
Of course it might be a good idea to talk to her now, if you feel comfortable, but if you really are worried about her posting things on facebook or telling people, then I understand why you might not want to say anything. Good luck - it'll get better :)