what sort of bullying are you experiencing? Physical? Verbal? It is natural to want to get some distance from siblings so you can have your own life and independence, even when you do get along with each other, but these might be issues that you need to confront rather than run away from.
Is your sibling aware that you're feeling this way? To them it might be jokes, but to you its a lot deeper. It's bad enough hearing verbal abuse in public or in school or at work, but when you're at home with family, you want to feel safe and accepted. Home is the last place you want to feel isolated and cut off from people. You need healthy communication with your family. So if you can, stand up to your sibling and let him or her know that you can't tolerate the things they are saying any more because it is impacting other parts of your life and making it difficult for you to find happiness with other people. You deserve better than that. I get along fine with my siblings now, but they did make a lot of mean gay jokes when I was younger, and it made me feel like I didn't deserve to be happy and have friends. It ruined my confidence and messed me up for quite a long time. What sorts of things is your sibling saying?