my boyfriends ex?

So I’ve been dating this guy for a month now. He’s amazing! He’s super caring, honest, respectful, and treats me perfectly.

However, he still is friends with his ex. They dated for a year about 4 years ago and he told me he still has some feelings for her. They still talk, they FaceTime sometimes, and they snapchat. She is a TERRIBLE person. She cheats on all of her boyfriends (including mine when they were still dating), does a bunch of drugs, sleeps around, and is just a mean person. She is also the type to go after guys with girlfriends, and is perfectly okay with being the side chick.

What bothers me the most is that he hasn’t told her that we are dating yet. He keeps telling me he will, but he hasn’t. Last night, he sent her a picture of me and she got really jealous. He says that he’s scared to tell her because she’ll freak out and try to ruin our relationship. She used to tell him all the time that “she will be his girl someday” and it BUGS me.

He loves that I’m okay with them talking, and keeps assuring me that they are just friends. I’ve told him that it bothers me, but that I trust him. Idk what to do. I don’t want to say something too harsh because I don’t want him to think I’m crazy or scare him away. I really like this boy and I feel like I want to be with him for a long time. We’re so alike in every aspect.

I need help because I really don’t know what to do about this.