Istanbul still sees decent construction activity both downtown (where old buildings are torn down and replaced with taller ones) and in the suburbs (the city got so big that we inadvertently say "I'm going to Istanbul" when we really mean the city centre).
So in theory, as the construction teams dig not only a wider area but also deeper, they should discover more and more historical artifacts, right?..
In Turkey, law says you can own the land but not any natural resources found underneath. So if word gets out that you've stumbled upon oil, stuff of archeological importance, gold etc. etc. while digging to lay foundations, your construction project is halted until the authorities decide what to do with your discovery. You may even be forced to sell your property back for its fair market value (ie. you get a decent offer but you can't refuse it -- you can just sue to get the offer increased)
You probably see where this is going: The sytem as it is incentivizes any artifacts found during private construction activities to just vaporize. Tragic, but I don't know what could be done about this. Only stuff found on state-sponsored projects seem to see the light of day, and of those that do, only ones that are useful to politicians are made part of the daily government propaganda, so you don't really hear about others unless you make an effort.
With this said, as a counterpoint maybe, they found a decent amount of stuff from non-Turkish / non-islamic periods during the recent boring operations for the newest subway line and they should be on display somewhere.