The Syrian Observatory for Human Rights said half the city of Afrin was captured but that there were still some "violent clashes."
Turkish officials also told the U.S. to withdraw its forces from Manbij, a border area in Syria.
Turkey considers the U.S.-backed Kurdish fighters to be terrorists.
Tillerson gave the policy speech at Stanford University.
Turkey's president is accusing the U.S. of supporting terrorism, thanks to U.S. plans to back a Kurdish force on Turkey's border.
Former national security adviser Michael Flynn allegedly plotted to abduct a Turkish cleric, according to a Wall Street Journal report.
The arrest of a Turkish employee of the U.S consulate in Turkey has led both countries to temporarily stop issuing new visas.
Thousands marched in protest of the Turkish president's crackdown on dissent since a failed coup last year.
Amnesty International said seven activists and the director of its branch in Turkey were arrested "blatantly without cause."
Turkey's Parliament offered the diplomatically isolated nation of Qatar a show of support by speeding up a bill to send troops to the country.
The country's new constitution allows the president to join political parties again, so Erdogan became the head of Parliament's largest party.
Turkish security officials were involved in Tuesday's assault of protesters against…