Okay soo I guess you should talk with your mom about it. Say to her that you want to wear bikinis not because you want to look like a "slut" (I dont know if your mom thinks that way) but because you like how they look. (Or whatever the reason is why you want to wear them) maybe tell her that you feel more comfortable wearing them (if you do) because people won't laugh at you etc.
Discuss with her and ask her why she doesn't let you wear them.
If she still doesn't want to let you wear them, maybe you should consider telling her what happened with your friends wen you were buying swimsuits.
Just try to stay calm and have a good reasons why you want to wear bikinis

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Hope that helped even a little :)