ISTANBUL — Turkish President Recep Tayyip Erdogan vowed Monday to keep up an offensive against U.S.-backed Kurdish militias in Syria, rejecting American calls for restraint and boasting of a deal with Russia to press ahead with the assault.

Erdogan's defiant message underscored the deepening rift between the NATO allies over the Kurdish militias and signaled a possible escalation of the latest tensions in Syria's seven-year conflict.

Turkey sees the Syrian Kurdish fighters as linked to insurgents fighting for Kurdish autonomy at home. Washington, meanwhile, has turned to the Syrian Kurds as a proxy force against the Islamic State and a bulwark against efforts by the extremists to reclaim territory.

Turkey on Saturday announced an air and ground offensive to rout the Kurdish People's Protection Units, or YPG, from Afrin, an enclave near the Turkish border. U.S. officials quickly called on Turkey to limit the scope and duration of the operation to avoid civilian casualties.

"We appreciate their right to defend themselves, but this is a tough situation where there are a lot of civilians mixed in," Secretary of State Rex Tillerson told reporters while traveling from London to Paris, according to a pool report.

"Turkey has legitimate concerns about terrorists crossing the border into Turkey and carrying out attacks," he said, adding that the United States has asked Turkey to "just try to be precise, try to limit your operation, try to show some restraint."

But Erdogan offered little suggestion that Turkey would scale back its offensive. "We are determined. Afrin will be sorted out. We will take no step back," he said at a meeting of business leaders in Turkey's capital, Ankara. Without elaborating, he said Turkey had reached an agreement with Russia — whose forces back Syrian President Bashar al-Assad — over the operation.

"America says the timing [of the operation] should be clear," Erdogan continued. "Well, was your timing in Afghanistan clear? Is your time in Iraq done?"

Syrian Kurdish officials said Monday that at least 13 civilians and three Kurdish fighters had been killed since the operation started. Turkey also deployed allied Syrian rebels to help in the fight.

It was unclear, however, how far Turkey or its proxy forces had advanced on Afrin or surrounding Kurdish areas. Turkish officials said this weekend that the goal was to create a "secure zone" along the border.

In Afrin, a spokesman for the YPG, Nouri Mahmoud, denounced Russia for apparently giving a green light to the Turkish attacks. "This is an unethical position from the Russian forces," he said at a news conference.

U.S. officials say the YPG militia played an essential role in ousting Islamic State militants from several areas of Syria.

"They have proven their effectiveness," Defense Secretary Jim Mattis told reporters Sunday en route to Southeast Asia.

"It has cost them thousands of casualties," he said. "But you have watched them, with the coalition support, shred [the] ISIS caliphate in Syria." ISIS is an acronym for the Islamic State.

Alex Horton in Jakarta, Indonesia, and Heba Habib in Stockholm contributed to this report.