BEIRUT, Lebanon — They seek to sweep away borders and establish a stateless society. Their female fighters struggle and die beside male comrades. Their leftist, anti-Islamist image has attracted American and European volunteers.

The Kurdish fighters who are battling the Islamic State jihadists in Syria are regarded by the United States as its most reliable partners there. But to Turkey, a NATO ally of the United States, these Kurds are terrorists.

The Kurdish group, known as the People’s Protection Units, or Y.P.G., is now facing an escalating battle with Turkish forces in northwestern Syria, complicating American policy.

The group has deep ties to the Kurdistan Workers’ Party, also known as the P.K.K. Both Turkey and the United States consider the P.K.K. to be a terrorist organization for its violent separatist movement inside Turkey.