You’re just seeing what he’s doing online and that information is freely available to the world.
Your motivations for checking up on this are worth looking at, though, because it gives me the feeling that either something inside you feels like you don’t quite trust this guy or that you don’t trust the relationship you’re in to have trust as a quality (and so you’re always checking and testing because you don’t have that trust to begin with…

It’s reasonable to interpret that as meaning you’ve agreed to not date anyone or sleep with anyone else, but I want to ask: when you agreed to be exclusive, how did this come about?
How clear was his side of the agreement to being committed?
At this moment, you are reading into the situation negatively, assuming that he has bad motives or could be playing you.
The fact remains though that you don’t know, but hopefully having a conversation with him will bring you closer to knowing. It is true there are dishonest people in the world and it’s perfectly reasonable and healthy to be suspicious when you get the sense that you’re with someone who’s lying to you. However, there are times where we ourselves are paranoid…
I can tell you from personal experience that this was a skill I needed to learn.
In my past, there have been times when even though the relationship was good, my insecurity would eat away at me.
I would think things like, “Well, things seem good, but what if she’s doing something behind my back and playing me for a fool, etc.
etc.” This type of thing has a lot more to do with our own personal insecurities and not so much with what the other person is or isn’t doing.
As such, some of the comments (which I have preserved) bring up points that I have since addressed in this revision.
Right off the top, you mentioned that you and he have agreed to be exclusive.
A problem with suspicion and snooping is: the more you fear and suspect, the more that fear and suspicion eats away at you and creates more fears and suspicions!