First to mention that Kurds (just like Armenians and Assyrians) are one of the ancient people in the east of Anatolia.
Kurds had mostly converted to Sunni Islam during Arab conquest, and later the immigrant Turks to Anatolia were also Sunni Muslims.
This was the point that Kurds - unlike other people of the region - became friends with Turks, but always kept their "Kurdishness", I mean that they didn't lose their race, language and Kurdish culture. They have kept it until today.
But why groups of them sided with Turks during the Armenian Genocide?
Because they were never the target of "pan-Turkism" during the Ottoman Empire, and they were not threated.
While at the end of 19'th and beginning of 20'th century all the nations under Ottoman Empire (Armenians, Greeks, Arabs and Slavic people) were fighting for independence, Kurds were still with Turks.
Also to mention that many Armenians during the genocide were survived by Kurds.
Up to this point, Kurds and Turks are friends.
BUT in the "Modern Turkey" things changed...
This time, the only non-Turk people of Turkey were Kurds. All Christian people of Turkey were wiped out as much as possible before, during the Ottoman Empire.
For this, Turkish government started his new program against Kurds.
For example they said that "Kurds are a kind of Turks"!!!
They called them "mountain Turks" and so... ( Denial of Kurdish identity!)
While Kurds are from Indo-Iranian branch, they were told mountain Turks.
(Newly some of Turkish officials have said that Kurds have Turkmen root!!!)
Kurdish language was forbidden and they couldn't learn their mother tongue.
Actions like these, didn't remain without re-action from the Kurds, from this point the Kurdish-Turkish relations started to change, and they started to think for independence.
It was the real problem among them, but also some other reasons such as PKK terrorism made the problems deeper.
Maybe you're right, in recent time PKK terrorism is the main problem between Turks and Kurds, but how it started?
It started as an answer to anti-Kurdish actions of Turkish government.
Of course I condemn PKK and terrorism, but everything has got a root...
I made it as short as possible