ISTANBUL — Turkey intensified its offensive against a Kurdish enclave of Syria on Monday, advancing troops in a ground assault against American-allied militias there, in a clash of interests between the two NATO allies.

Turkish forces, backed by warplanes and artillery, captured high ground and three villages near Afrin, a city controlled by a Kurdish militia that Turkey regards as a terrorist threat.

The Turkish incursion, coming over protests from Washington, not only underscores the Trump administration’s lack of influence with Ankara but promises to complicate relations with the Kurds, who have provided the ground troops for the United States-led fight against the Islamic State militant group, often called ISIS or ISIL.

The problem of Washington allying with the Kurds, who Turkey considers terrorists and a threat to its territorial sovereignty, could be overlooked as long as ISIS remained a threat. But the group is now in retreat, leaving the administration searching for a way to maintain relations with the Kurdish groups without alienating Ankara.