BAGHDAD — In the days after reaching an agreement with the United States to combat mayhem in Syria and Iraq, Turkey said it had vaulted itself into the battle against extremists menacing Turkish security.

But the extremists the Turks have in mind are not just members of the Islamic State. Instead, as has become increasingly clear this week, Turkey is at least as focused on crushing the Kurdish militants it has struggled to contain for many years.

President Recep Tayyip Erdogan of Turkey increased tensions with Kurdish militants on Tuesday, telling reporters it was impossible to continue a two-year-old peace process with them.

He spoke as Turkey, a NATO member, convened an emergency summit meeting of the alliance, suggesting that Kurdish separatists were at least as big a threat to Turkey’s part of the world as the fighters of the Islamic State. The Turks got a NATO pledge to battle both groups.