Help?

I'm bisexual as well, leaning towards a higher attraction to men. So, I have a very similar situation as yours.
I would argue you're normal, but it really depends on the definition of what normal we're going at. In this case, it probably just means being heterosexual because there's a larger population of heterosexuality. Normality is not always possible to conform to, as such with sexual orientation.
....Um, lets not dwell on normalcy too much.
The simple fact is, you can't satisfy everyone. If you're normal, you will still not satisfy everyone. Someone will always dislike you. That is...considered normal. Biphobia is a huge issue. The fact that your family accepts you makes you somewhat luckier than most LGBT people. I would try to find friends that are also in the same situation as you, that way you can share experiences among each other. Do not ignore your sexual orientation, accept it. Ignoring your true self brings consequences and inner pain. I can't really prove that, but you can trust me on that. I would advise you not to self-harm. You can still lead a good life. Focus on your talents, and pain will slowly flow away.

Being normal is being who you are. Society doesn't need to like you, only you do. And whenever you feel like self harming, write to me and I will chat with you when I can. Also, your heart chooses who you love so let it. If it so happens to be a girl that is fine, as long as that is what you want. All of us just want you to be happy!:)