Hopefully this won't get more rambly than you can bear, and I'll try and keep things succinct. You're welcome to dig in more with any parts of this you want if you want to talk about them or want me to say more about them.
1) It's important for your personal development for you to have age-appropriate parenting. I want to toss you a couple links to look at to get a sense of what that can mean and involve:
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http://www.cdc.gov/ncbddd/childdevelopm ... cence.html
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https://www.healthychildren.org/English ... rents.aspx
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http://www.pamf.org/parenting-teens/hea ... .html#Tips for Parents
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http://lifehacker.com/how-to-give-your- ... 1638140713
I just feel like it's important for you to get some perspective on what healthy, developmentally appropriate parenting of someone your age can look like, because it sounds like it isn't happening in some big ways. Having a sense of what is appropriate and sound, and widely understood to be healthy is something I think might help you start to advocate for yourself better, and also at least know in your own mind when your parents aren't really doing this, so that you don't basically just get on board with what isn't healthy that they are rationalizing.
2) I would suggest you start taking some of the steps you can to gain -- or at least seek -- more autonomy and independence. It's very hard to have a sense of ourselves, and to not just kind of think or feel what others tell us we should -- even if those things are doing us harm in any way, or aren't what we think or feel -- without at least TRYING to basically stand up for yourself as your own person, and a person who is entitled to be treated as someone at the developmental level of life that they're at. One example is something I brought up: telling a toddler they can't bake because ovens are hot and dangerous is one thing: telling a teen that isn't at all age-appropriate.
So, I'd encourage you to see if you can't start asking for some things that are sound and appropriate like:
• Having the freedom to go some places on your own or with friends, like the library (as you brought up that's not allowed), to their houses, to the park, hanging out afterschool, or to something like a volunteering job of some kind. You can say you would like to do any of that, then ask for their permission. If they refuse, you can next ask what you can do to set a foundation for that changing soon. In other words, try to negotiate by asking what they need from you to give you -- and I would put it this way -- the ability to do things that are age-appropriate for you, and that most other people your age are supported by their parents in doing.
• I'd do the same with other things, like -- assuming you have any interest in this -- baking, like you brought up as something you are restricted from. If there are other kinds of basic life-skills things like that, I'd go for those as well. And just like with the above, if they refuse, I'd take the next step of asking to negotiate.
• If, with any of this, they just still refuse or won't even have discussions with you, I'd leave it for now, but keep bringing it back up. Like, if you ask about going to the library alone, and they say no and won't discuss it, next month, you try again, and you keep trying. Even if these things don't net in you being given permission, I feel like you can benefit just from trying to advocate for yourself and trying to get room to be your own person who is growing into an adult. I think that it will help you to start to get and better develop a sense of your self that's more separate from them, and thus, also more resilient to things like sexual shaming.
3) Make the most of your time away from home at school. In other words, while you're NOT at home, be sure you're really doing all you can to utilize the freedoms you're given at school. Get offline more, for crying out loud.
It's easy to stay sheltered if your primary environments are a family that shelters you and online. Get more real life into your life, as much as you can. I think it can help with your self-esteem, and again, help you with resilience: it seems like something you struggle with a lot are what others think, and it's easier for that to hit us harder when we're not doing all we can to just live our lives fully and enjoy and explore them as much as we can, if you follow me.
4) Do what you can to reach out more so you can have a larger social circle and a better support circle, where people have a range of beliefs and opinions. For instance, maybe you need to try reaching out to your siblings more for support, to make some new friends who are very different than you, and to look into what kinds of helping resources are within reach to you that might benefit you: things like local community centers, teen centers, a school nurse (since it seems like you don't have a healthcare provider you see often), or even to ask your school to give you a list of what resources they have for teens who want counseling (if they don't have a school counselor, they should have a list like that).
5) Start a journal where you are exploring what YOU feel, think and believe, and what YOU want for your life. In other words, I feel like more focus on who you are, as your own person, and what you want, could help you out. Just the practice of doing that usually helps us clarify all that for ourselves, and strengthen our resolve to go for who we want to be and what we want our lives to be like. Obviously, you can't magic all of that all at once, but just starting on that process should be helpful and start you on a route to some positive changes, including feeling better about yourself and common, healthy parts of life, like sexuality.
I do want to say one last thing in this post, which is that I have this niggling feeling I am missing something big in your family or with you that might be part of all of this. A developmental disability, abuse or mental illness in your immediate or extended family, I don't know -- and those are just a couple options off the top of my head -- but I feel like SOMETHING is missing here. Maybe it's not, and if something big is missing because you don't want to tell me about it, that's okay, you don't have to. But I do just want to leave the door open to say that if there is something major you think of later, or that you don't want to share now but do later, I can certainly revise any of this and we can keep talking, regardless. I just wanted to put that out there because in the event anything I have suggested strikes you as earnestly unsafe for you in any way I might be missing, please go with what you know is safe and sound for you. You're more the expert of you and your life than someone like myself can ever be.