Let's face it – you have been hiding part of yourself for years from someone who thought they knew you and from someone whom you love. You have good reasons for that – give them the reasons, help them understand. One bite at a time. When they bring up the topic comes up, tell them a short story of a specific incidence from you past where you had feeling of being a girl. Something like "When I was six, I was jealous of a dress someone was wearing." I hid this away because I was ashamed/didn't want to hurt you/didn't want public exposure etc. (pick one). We should have talked then. I realize that now that I was wrong/mistaken/not the best thing to do. I just wanted to be a normal boy. I am sorry." You get four or five at most sentences to give the incidence, how you felt, why you did not say anything then and apologize for not speaking earlier. Every time this comes up, you do the same thing with a different incident. And don't make them recent – this gets you out of the "It's your friends/ the internet/ the counselor." Use different descriptions and different words. Over and over again. No hugs unless they start it. Tears are good.
Also, every now and then, as you do this a few times, relate something that happened as a result of your feeling that was good. For me (JLT1), as an example, my hiding from what I am led me to tremendous success in my career. My hanging around my sister because she dressed so well and I was jealous meant I got to know one of her boyfriends and from there, played and enjoyed football. It has to be something that brings up mutually shared and fond memories. This is the opposite of above – the longer and more in-depth, the better. You are firming up the relationship, making it stronger.
You love them – forgive everything NOW. They do not understand, they are angry and lashing out. It's the relationship that is important, the apologies come from you first, and then the apologies will come from them. Swallow the pride, the hurt and your own (justifiable) anger for the sake of the relationship. I'm NOT saying this is right in anyway, it is only what works.
As far as I am aware, this is the only way to do it. It carries a risk, they may feel you lied to them forever and you're not the person they thought. However, this is the process your psychologist used with you to determine that you are transgender. Your helping your family though the same process.