I was a lot older than you when I converted and it was still a difficult thing to tell my family. I come from the area of southwest Virginia, there near the border to KY, TN, and WV. Dickenson county. My parents were very understanding. Some of my cousins, aunts and uncles however do not speak to me anymore.
My only advice is this: when you are so sure that you are following God in this choice in your life that you are ready to convert, even if it costs you friendships and relationships, then it’s time to tell them. If your family truly loves you they will support you in your decision. My mother doesn’t understand, but she’s there for me. Even though she was concerned for me becoming Catholic, the other day she bought a rosary for my 4 year old and is making it by hand for her. She loves me enough to support me even if she doesn’t understand.
Pray about it. Find a way to tell them, and emphasize that it is God leading you to do this. Make sure that you don’t make it seem like you are leaving them, but that you are simply following God in your heart. Sometimes they feel rejected, as if you are not only rejecting your religion, but your family also. It’s important to make sure that they feel your love for them in your life, so they know you are simply acting out of love for God AND them.