Is he cheating?

I was at my boyfriend house and I went onto his laptop to search something up but I accidentally closed the website. So I tried looking for the same website through his search history. Only to find out he's finding interest in someone else.. (one of my best friends) They used to talk a year ago and my friend told me that she feels uncomfortable about how he's been complimenting her. I've confronted him saying i'm not comfortable with him talking to her. He apologised and stopped for awhile. It's been 2 years now and she tells me he occassionally texts her asking random questions like, "whens this assignment due?" blah blah. I shrug it off since it's not such a big deal and I trust her.. but recently I told him that she's had sex with her bf for the first time and told me about it. Now i'm worried about what i've found in his search history..

I'm really not sure what I should do. Him and I have always had trust issues with him doing things with other girls..

and we've gotten over that.

Few months down the line.

He can't let go of this one particular girl.. I don't know how to confront him about it since I invaded his privacy but I don't how to go about it.

I would usually be crying right now, feeling hopeless but now i'm just concerned.

Should I just break things off officially and never take him back?

or am i being immature..