The Kurdistan Regional Government and the Kurdish Democratic Party (the coalition party which runs northwestern Kurdistan) have protested the incursion but will most likely will do nothing beyond that. Kurdistan's Peshmerga fighters have moved in to keep the PKK from trying to retreat south. Any southern withdrawal would likely entice the Turks to move in deeper. Turkey says there is no timetable for withdrawal but the United States is pressing for one as soon as possible.
The fight has been hampered by the mountainous terrain and bad weather. Turkish commandoes are fighting in waste high snow and freezing cold temperatures. These conditions will likely deny Turkey the "crushing blow" they seek against their long time rival.
Terrorist and other groups in mere presence in Iraq create the risk of other incursions. The Islamic Communist cult People's Mujahedin of Iran (MEK) in the past killed major leaders of Iran and remain a target of Iran's wrath. Mahdi Army Special Groups and Hezbollah are active in Iraq. A big attack by either of them (though they attack Coalition forces daily) could cause Coalition forces to target Iran.