They are not Persian (although in the same linguistic family) or Turkic, they are descended primarily from Iranic tribes (mostly Median with some Alani) with substantial Hurrian and Assyrian admixture and more recent and small influx from Turks and Arabs.
Kurds within their own populace are quite diverse, with several linguistic branches that have separated due to their lack of a uniform system of writing since Median times. And some peoples such as Zazaki and Lors are incorrectly labelled as Kurds by ignorants or Kurdish nationalists.
The Gorani (Hawramani) Kurdish language is considered a Middle Iranian language and is very close to Sassanid Persian, Zazaki language is also considered to be a Middle Iranian Language and is said to be directly descended from Parthian, just like Sangesari in Iran.
I encourage you to perform a forum search of "Kurd", you will find many topics discussing them. I have Fayli Kord, Lori and Persian blood.
Edited by Zagros