Turkish forces began a long-anticipated cross-border assault on Wednesday against the Syrian Democratic Forces, a Kurdish-led militia, in northeastern Syria.

The dispute between Turkey and the Kurds has deep roots in regional power dynamics that have created a tangled web of interests. Further complicating the picture is the fact that the United States is an ally of both Turkey and the S.D.F., as the militia is known.

President Recep Tayyip Erdogan of Turkey said the goal of the incursion was “to destroy the terror corridor” that he said Kurdish forces were trying to establish on his country’s southern border, and to bring peace to the region.

Leaders of the S.D.F. and others in the region say the strikes are putting civilians at risk, and warned of an imminent humanitarian crisis. Kurdish groups on the ground shared photographs and videos of people fleeing villages as smoke rose from the site of strikes.