"We have to discuss how to move forward together." - To me, this sounds more like actually moving forward together rather than separately. But I agree, it is not straightforward.

Maybe she wants you to sit with this discomfort for a while, to think about how an obsession like this can affect your relationships, instead of just saying basically it's fine and there won't be negative consequences. In a way, if you have an addiction-like pattern, it may not be bad to be forced to tolerate negative feelings and anxiety rather than just getting an instant, superficial gratification that all is okay. It can also be that the T is upset about it and gives you a bit of passive aggressive reaction. In any case, I don't think anyone can really tell what will happen. I would definitely prepare for discussing what makes you do all the searching though, what this preoccupation means to you. Again, I very much believe there is nothing objectively wrong about looking for information online about anyone as long as you don't break into anything, do illegal things, or use fake identities to fool people. But it can still affect relationships, don't know what your experiences have been with other people. This is why I would never tell anyone, I think it is the telling that's actually intrusive, not the looking on the public web. But maybe it's good material for therapy, I don't know. I actually always have a hard time believing that many Ts don't look up clients they find interesting in a similar way, I know I would have a hard time inhibiting myself if I am curious.