This is the best advice in this thread.
If it were me, in a very confident no BS tone, I would talk with her and open with something like:
"Sally, I've been thinking about your "dates" (and they are, don't let her use her conniving, "Oh, he's just a friend" BS that you'll almost certainly get from her--BIGGEST cop out in the female playbook) with the guys you previously met on dating sites. I'm a bit confused as to why you feel compelled to meet them, again, for what appears to be another date when you and I are dating exclusively. It may be ok with some guys, but it's not ok with me. So feel free to go out on the dates and have a good time, but if that's your choice, I don't think we should be exclusive any longer. I'd still like to see you, but just know, like you, I'll be seeing other people too."
PS- A girl I was dating a number of years ago pulled the same crap with a guy at work she was incandescently talking about and having what I call an emotional affair with. I went over to her place unannounced, said my piece and walked out. His name came up 95% less and her IL went up. That was within our first year of dating. We ended up staying together for 9 years and a lot of it was due to the standards I set early on. I was tested on many occasions and usually used the "Some guys might be ok, with it but I'm not" angle. It always worked.