My boyfriend flirted with another girl. How should I move forward?

I know for a fact that I will be called stupid for not breaking up with him but please hear me out because I need help. Also this might be long.
Recently I talked to my therapist about having trust issues and that I worry about some of the girls he talks to. I understand many are platonic but he’s a bit secretive about one of his female friends. I was told to ask him to show me their texts but not to force it. So I told him, in a calm way, “I’m feeling nervous can you please show me your texts with *her name*. If you don’t want me to look at something, I won’t look”. As I was looking through the texts, it was mostly normal but there were some moments that bothered me. One of them is when she sent a video of date ideas in the area where he lives. He responded with “Are you planning on taking me on a date? 🤔” to which she responded “your girlfriend wouldn’t approve”. Another text I saw was with a meme of an animal fighting another animal and he said “my girlfriend when I talk to you”. Also she is living in another country right now and he kept asking her when she was coming back. I cried to him asking why he would say these things and he tried to argue with me, bringing up random stuff like I take up too much of his time and I had an attitude that day. The next day, he was very affectionate and kept saying he loves me and he knows he did wrong and he won’t do it again. We’ve been together for quite some time so I don’t want to make an impulsive decision. The only reason I didn’t break up with him immediately is because it wasn’t fully cheating and it could have been harmless. I just want to get other perspectives. Do men really mean it when they say they’ll change in these situations? How can I heal from this and is it even possible to get over it? I’m absolutely heartbroken.

My boyfriend flirted with another girl. How should I move forward?
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