Seriously, modern Turkey people especially one who live near and around Istanbul and west Coast is lot look like Southern European to me.
Yes well looking at history, its not hard to see why. I bet even some Hittite DNA markers stays among anatolians.
I have lot of dravidian (ASI) genetics in me too (alongside "indo-aryan" aka ANI), the former is simply the people that have long lived in the area I am from and over time ANI migrations and contact arrived....and some level of fusion or replacement w.r.t culture/language can happen depending on scenario (and this influence your identity w.r.t nation etc)....but it wont alter underlying population genetics quite so much.
Thus its why you see not much genetic difference between lot of Turks and say Greeks....historically all one need to do is read say the Illiad....and look where Troy was. Invasions and migrations all overlay their bit downstream, but its not so drastic change overall.