The suggestion contains a false premise. A lot of Anatolian Turks including myself do have epicanthic fold, but of course, it’s less prominent than what East Asians have. Have a careful look at this photo collection. It contains hundreds of photos many of which state the person’s name and province/city of origin in the file name.
The existence of epicanthic fold, prominent cheekbones and flat/petite noses in Anatolian Turks is explained by the fact that Anatolian Turks have a significant percentage of “Mongoloid” admixture.
Maternal “Mongoloid” ancestry: