It sounds like he may need some professional help to work through some things in his past. That’s not something you can help him with. 

I also think he’s trying to play the victim by saying you’re insecurities are making you not supportive of him. Which is just kinda shitty of him. His behavior, the constant lying, would make anybody insecure and paranoid. He clearly isn’t able to move on from his ex. Which, if it were a healthy relationship (they exist), wouldn’t necessarily be a problem, but the things he’s doing is disrespectful to your relationship. 

It’s a good rule of thumb that, if you can’t do something (with somebody else) openly in front of your partner, or with their knowledge/consent, you probably shouldn’t be doing that thing.