"Sen Ayasofya'yı biliyorsun" means you know about it or have heard of it.
"Sen Ayasofya'yı iyi biliyorsun" means you know quite a few facts about it.
"Sen Ayasofya'yı biliyor musun?" could EITHER be asking if you've heard of it or if you have knowledge of it, but more likely the former.
If you wanted to specify asking whether I have a good knowledge of it, you could ask "Sen Ayasofya'yı iyi bilir misin?" or "iyi biliyor musun?"
"Sen Ayasofya'yı duydun mu hiç?" means "have you ever heard of the Hagia Sophia before?" Like, heard it referenced.
Why is it that certain verbs in Turkish seem to be inclined to use the continuous tense? Bilmek, Sevmek, I always seem to see them as biliyorum, seviyorum etc. Is there anything specific to their type of verbs, is it just a casualty of the language? It's like this sentence is saying "You are knowing Hagia Sophia", right?