I think that normal is an overused/misused word. We're all so different... who is to say what normal is? You are attracted to men as well as women from what you're saying. Does that make you abnormal? It makes you different from your family apparently. In their views you may be weird. A lot of us are different from our families... those differences become more pronounced and defined as we grow up. We develop our own views and outlooks. We begin to become who we are.
I don't think you should worry too much about the what-ifs reactions of others right now. Find out who you are and try to become comfortable with it. If you do turn out to be bisexual that isn't going to mean you're a bad person. It's just going to mean you're bisexual.