Not all Kurds want the same thing.
There is a Kurdish terrorist organization called the Kurdish Workers' Party or PKK. But of course most Kurds do not belong to it.
Kurds generally get along well with the US because their values are similar. For instance, many religions live together in harmony. And instead of hoping some outside agency will help them, they make an effort to help themselves. They are quite effective, within their own range, in combatting DAESH and other terrorist groups.
Kurds live in Turkey, Armenia, and Iran as well as Iraq, where these countries border each other. Since they have their own culture different from these other countries, many think it would be better if they were organized into a single country.
The Kurds were quite pleased when the UN invaded Saddam's Iraq, and got along much better with the new democratic government that was installed. But a lot of things have changed in Iraq since then.