the TurkISH people are genetically far closer to Greeks, Armenians and Iranians than they are to the TurkiC peoples (by some studies only about 5% of their genes are East Asian).
The oldest known Turkic inscription–thus, our first known use of the Turkic language–comes from the Orkhon Valley, which is located smack dab in the middle of Mongolia -- some linguists propose that the Mongolian family and the Turkic family might be related to each other within a broader “Altaic” language family, which makes plenty of geographic sense, but might not make true linguistic sense.
This (difference between genetics and languages) isn't unusual - the English, despite speaking a “Germanic” language, only about 10% German by ancestry (more or less;)
But back to the Turks
We can see the "bleed" of different ethnicities merging.