An independent Kurdistan was actually planned for and included in the Treaty of Sèvres. However, the successes of Mustafa Kemal Atatürk's newly organised Turkish republic meant that the Turks were able to take control of the area earmarked as being Kurdistan.

The western allies were not interested in another war with Turkey, and the Treaty of Sèvres was abandoned in favour of the new Treaty of Lausanne which set the new borders of Turkey, without any Kurdish state.

There's a separate issue from the top comment which is that Kurdish uprisings in northern Iraq began almost immediately after the end of the war. At the time, the British wanted to stay in Iraq permanently (or thought they did at the time) and knew that if the predominantly Kurdish Mosul province was a part of Iraq (as opposed to Syria, as had been originally planned in Sykes-Picot) that the Hashemite government would be reliant upon British airpower indefinitely.

Faisal and his clique of Iraqi Arab nationalists wanted Mosul as well, not just for its oil but because they wanted Iraq to be as large possible.

Overall the idea of an Iraq consisting of Basra, Baghdad and Mosul provinces was that Sunnis (including Kurds) would counterbalance Shia, while Arabs (both Shia and Sunni) would counterbalance Kurds. Demographically there would be just enough Sunnis to justify Hashemite rule and it would be just hard enough to govern to keep the British involved.

With the great depression, however, the British were forced to withdraw early.