I don’t think the reactions of your parents and relatives to counseling is your primary problem. I’m going to venture the guess that their opinion matters to you too much. In your case, it makes sense to me that it would be so. Your relationships with people you care about have been disrupted many times. It’s understandable that you are overly concerned about alienating anyone.
But — It is normal and appropriate at this stage of life for you to be forming and acting on your own opinions — with one caveat. Part of being an adult is knowing how to hold onto what you know is right for you without attacking or diminishing people who care about you but who have different opinions.
As you probably know, the best predictor of effectiveness of therapy is a positive relationship with the therapist. You do need to find a therapist who feels like a right “fit” for you. Then calmly and clearly make your request and present what you know about the therapist you have chosen. There is no need to defend yourself. There is not need to try to convince your relatives that you are making a reasonable request. Simply state that it is something you need and that you would would appreciate their love and support.
You are 18 – soon to launch into adulthood. Hopefully you are preparing to either get a job or go for further schooling as you work toward becoming independent. Because of your history, a unique challenge for you and your relatives is to celebrate that independence and not see it as a threat to the relationships.
I wish you well.