@georgie78,
Are you over 18? If so, you have a right to privacy from your doctor.
I come from a similar background, and I think I understand how you are feeling. Religious communities place a lot of shame on masturbation or any form of sexuality, and this do some weird things to your mind. I am not a doctor, but I find it highly unlikely that you would do any lasting physical damage from masturbation.
Hopefully you have been given the message that masturbation is normal for most men and is part of healthy sexuality.
I just re-read your post; that you are over 19. This means that it is illegal for your doctor to discuss anything you tell him with your family (assuming you are in the US). You can talk about this with a doctor, and they they are prohibited by law from discussing it with your family. If you don't trust your current doctor, you can go get another one.
Most doctors will understand this situation (it isn't uncommon) and will give you the advice you need.