There has been fear that decades of enmity between the two could spill over into all-out war
Ankara ordered a cross-border operation into Syria on October 9 because it said it wanted to create a security cordon free of Kurdish armed groups that it considers to be terrorists, linked to Kurdish rebels inside Turkey.
The long-planned operation started after Trump announced the exit of the small, but politically significant US military force which had until then been closely allied with the Kurds.
Trump said he didn't want the US troops caught in the middle of a Turkish-Kurdish war.
Accused both by Republicans and Democrats of abandoning the Kurds, Trump imposed sanctions on Turkey on October 14 and sent a delegation to persuade Turkish President Recep Tayyip Erdogan to order a brief ceasefire.
The first Russian patrol in northern Syria got underway on Wednesday, the defense ministry in Moscow announced.
Critics of Trump say he has caved in to Turkey and been outplayed by Russia.
"It is unthinkable that Turkey would not suffer consequences for malevolent behavior which was contrary to the interests of the United States and our friends," tweeted Republican Senator Mitt Romney.
Trump said that a "small number" of US soldiers would remain nearby, but purely to guard oil facilities.
His central message was clear: that the United States has no business in Syria and that there was never any question of trying to stop NATO member Turkey from carrying out its invasion.
"We have spent $8 trillion on wars in the Middle East, never really wanting to win those wars," Trump said.
The Turkish-occupied area was calm, the Syrian Observatory for Human Rights, a British-based war monitor, said on Wednesday.
However, explosions hit various parts of Syria's northeast, including the Kurdish-majority city of Qamishli and the town of Suluk in the Tal Abyad region near Turkey's border.
The car bomb in Qamishli did not cause casualties, according to the Observatory. But the blast in Suluk, controlled by Ankara's rebel proxies, killed three fighters, a rebel commander there told AFP.
It was unclear who was responsible for the blasts.
Kurds staged angry demonstrations Wednesday in their de facto capital Qamishli.
"This deal serves the interests of foreign powers and not the interests of the people," said official Talaat Youndes.
"Turkey's objective is to kill, displace and occupy the Kurds."
There has been fear that decades of enmity between the two could spill over into all-out war
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The Ministry of Foreign Affairs reiterated its call on the United Nations and the UN Security Council to fulfill their responsibilities
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Dressed in all black, the man was seen directing traffic for nearly six hours
Four others to drive away a luxury vehicle each