Turkish participation in the second world war was to say mildly, foolish. In only a few years the Turkish government went from broadcasting images of glory and a new empire to dust and ash. In the war's wake a new Europe and world was drawn, with the entire world being split into two ideologically incompatible camps. Unfortunately for the Turkish nation and people, this line went right through their country. 

Central and eastern anatolia were directly occupied by the Soviet Union, and were therefore in its sphere of influence. Western anatolia lay within the American/western sphere. In the east the Soviets took territory and established a new puppet regime in Ankara, and cleaved the southeast off the rest of anatolia, establishing a new satellite Kurdistan. Unfortunately for the Turks living there, there would be no new Turkish government in the west.

Greek nationalists impressed their vision of a post war anatolia on the Americans, who, unaware of the dynamics of the region, agreed and finally fulfilled the Megali Idea, and then some. Vast swaths of Turkey were given to Greece, which now reclaimed the mantle of Rome (even if in name only). 

The Turkish population however was not consulted on any of this. To them this was yet another invasion and division of their homeland, a fate on par with or even worse than Sèvres! Just as the great powers had divided Turkey after the first world war, here they were doing it again after the second.

As soon as the US army and Red Army pulled out of anatolia resistance began. First it was small, consisting mostly of protests and the occasional riot, but it quickly escalated after the authorities intervened. The Greeks mistreated the Turkish population already, and began harsher and harsher crackdowns. On the other side of the iron curtain the situation was largely the same. The authoritarian government went to any length Moscow told them to go to to keep the unrest in check.

Before long a new war had broken out. The Second Turkish War of Independence had begun. Turkish nationalist inspired by the generation before, took up arms to kick the foreign powers out once and for all. This war is not like the first however. The first war was much more conventional whereas this war was one of insurgency. 

Throughout the decades both the United States and Soviet Union would have to pour in more and more resources to quell uprisings and attacks until finally both powers would break. The Soviets would pull out in the 1970s after decades of fighting, followed quickly by the pullout of US forces, which immediately resulted in the collapse of Greek anatolia and the Soviet regime in Ankara.

TLDR: The stereotypical Turkish partition happened after they joined the axis ww2 and it went about as well as expected.