Northern Syria -- Turkey's President Recep Tayyip Erdogan said Tuesday that President Donald Trump had reassured him in a phone conversation that the U.S. was in the process of pulling its troops out of Syria, appearing to tamp down tension between the two nations which manifested itself just hours earlier in a testy exchange of tweets.

Turkey wants the remaining 2,000 American forces in Syria as part of the anti-ISIS coalition to come out, which would give the Turks free rein to launch offensive operations against Kurdish militia in the war-torn country's north. But many of those Kurdish fighters are U.S. allies who have been crucial to the fight against ISIS. The Trump administration has made guarantees for their security a precondition of the complete U.S. withdrawal from Syria.

Erdogan told his nation's lawmakers on Tuesday that he had told Mr. Trump the U.S.-allied YPG Kurdish militia, "tortures the groups in Syria that do not depend on them," and that his government had shared its evidence that America's allies are in fact terrorists with the White House.

But the Trump administration has not backed away from its insistence that the Kurds of the YPG be protected. Nor has it officially backed away from Mr. Trump's decision to pull U.S. troops out of Syria -- though that process is happening far slower than the president initially suggested.

In the meantime, CBS News correspondent Charlie D'Agata reports there's been an increase in the intensity of the battle against ISIS holdouts. The YPG-led Syrian Democratic Forces (SDF) are making the most of U.S. military support while they still have it.

Administration officials have stressed that the outcome of the battle against ISIS is not reliant on the physical presence of the roughly 2,000 U.S. forces in Syria, but D'Agata reports the Americans have played a vital role in the fight on the ground. They not only provide tactics, weapons and equipment, but crucially they're also directing airstrikes against ISIS targets.

D'Agata and his team witnessed that for themselves on board the American aircraft carrier USS John C. Stennis last week. From the waters of the Persian Gulf, wave after wave of F-18 fighter jets rocketed into the sky and headed for Syria, launching regular bombing raids against ISIS targets in Iraq and Syria.

While the partnership between U.S. forces and Kurdish fighters has proven successful, it has angered neighboring Turkey, which considers the YPG terrorists it has been at odds with for decades.

The Kurdish fighters who spoke to D'Agata in northern Syria are not only concerned that the withdrawal of U.S. troops could enable an ISIS comeback, but that Turkish forces will go on the attack the moment the last American soldier leaves.

Erdogan said Mr. Trump had also backed Turkey's long-held call for a Turkish-enforced safe zone, which would essentially create a 20-mile buffer along the Turkey-Syria border, but the Turkish president didn't explicitly say that zone would be a safe place for YPG fighters.

Spoke w/ President Erdogan of Turkey to advise where we stand on all matters including our last two weeks of success in fighting the remnants of ISIS, and 20 mile safe zone. Also spoke about economic development between the U.S. & Turkey - great potential to substantially expand! — Donald J. Trump (@realDonaldTrump) January 14, 2019

Mr. Trump confirmed the conversation in a tweet, saying he had discussed defeating ISIS, the buffer zone, and, "economic development between the U.S. & Turkey - great potential to substantially expand!"

For the moment anyway, SDF commanders tell CBS News they've seen no significant drawdown of American forces. That jibes with what U.S. military officials have told CBS News in Washington; that the withdrawal is underway, but so far only some pieces of military hardware -- no troops -- have left the country.

If anything, the Kurds say it feels more like a hurry-up offence, as they use the support of the U.S. military to wipe out the last of ISIS holdouts.