Is it normal my bf and i fight like this?

So my bf and I have been dating for just over a year and we started fighting because he txts this girl all the time. I looked through some messages and saw flirting he says it's not. We agreed to take a break get a taste of the single life and work our way back into the relationship, but we still have sex almost daily. He's also talking about an open relationship. I don't really want to but I don't want to lose him. We also fought over me flirting with another guy and I stopped. Is this whole situation normal?