The question of "race" is always troublesome. It stems largely from the perception that "race" is a reality in itself, but it is actually no more than a perception. There are many variations on human features, and what were classified as "the four races" are actually no more than four extremes based on features that stand out to us. Eye shape skin color, hair type, etc. In reality it´s a continuum, but when people migrate and come across people who look differently, then it´s (for better or worse) human nature to try and classify folks.
During Ottoman times, the major dividing line was religion, not race and ethnicity as we think of them today. They only became a real issue during the age when the world was dividing itsulf up into nation-states, and some of those states were trying to grab parts of the failing Empire. The reasons various people in the Ottoman Empire wanted to break away are more complex than at first glance, and beyond the scope of this post. But the upshot was that out of the Ottoman Empire arose a group of nation-states, and one of those nations were the people that considered themselves (or were considered by others because of language and religion) "Turkish." And one of the difficult issues the new state of Turkey faced was to define what a Turk was. Unfortunately there was a point where some tried to define what a Turk was in terms of physical characteristics. Clearly this caused lots of problems. For one thing, Atatürk would certainly not fit into any definition of Turkishness based on Central Asian ancestry. And it also left people out - consider all the people who identified as Turkish, who fought in the War of Independence, and then (in the eyes of some) were considered to be of "doubtful" Turkishness. Without naming names, people in some areas were more subject to such attitudes, and it´s in such areas where we see the most extreme touchiness and nationalism.
In the end, Turkish, Greek, Kurdish, we´re all mixed, and most of our nationality is a construct of our education and the culture we were brought up in. To get caught up in fights about ethnic or racial purity just reeks of ignorance and hurts everyone.
Although it looks like a few paragraphs but it is a very compact version of our history.. 