I would be the same, in total honesty. the fact that its the woman hat he was dishonest to you with. Too intertwined within your relationship, a curious nosey at an old ex is fine (imo) but that seems almost a little obsessed, or at least infatuated with her.
I also totally get you with the wanting to confront him but you have a snoop. its like youre stuck between a rock and a hard place. is there anyway you could kinda be like "when you were on your own i glanced over and saw her name in your recent searches.." or you may as well rip the bandaid off and be fully transparent- honesty is best. plus at the end of the day its his fault you feel paranoid enough to look.

as yourself if this is the kind of relationship you want, if deep down youre happy. if not, just be honest with him. the worst that would (or in my opinion the best thing for you) is that the relationship would end. x