In scenes all too familiar since the start of Syria's war in March 2011, civilians were seen abandoning their homes on Thursday, in vehicles or on foot with their belongings on their backs.
Esper called on the Turks to halt their operation, but he told reporters that he has no indication they are willing to do so.
Plumes of black smoke billowed Friday from the Syrian border town of Tal Abyad as Turkey continued bombarding the area in an offensive that was progressing "successfully as planned", the Turkish Defense Ministry said.
The third such Turkish operation since the start of the war in Syria has been met with fierce global condemnation over what many saw as the blatant betrayal of a faithful ally. Some made victory signs, shouting "Allahu akbar" (God is greatest) and waving Syrian rebel flags as they advanced towards Ras al Ain.
Some 120 km to the west, Turkish howitzers resumed shelling near the Syrian town of Tal Abyad, a witness said. It says it has killed hundreds of fighters since Wednesday.
Ankara has long opposed USA support for the YPG in the fight against the Islamic State extremist group, arguing the militia is a "terrorist" offshoot of Kurdish insurgents inside Turkey.
Kurds said they were resisting the assault.
Ankara launched "Operation Peace Spring" on Wednesday against the Syrian Kurdish People's Protection Units (YPG), which it considers a "terrorist" off-shoot of Kurdish insurgents in Turkey.
The SDF have been the main allies of US forces on the ground in the battle against ISIL since 2014.
Human Rights Watch (HRW) and other groups warned about everything from massive new flows of refugees to conditions for detained Islamic State (IS) fighters from a previous phase in Syria's chaotic civil war.
Trump condemns Iran's 'bloodlust' in scathing United Nations speech
I'm hoping there'll be a Trump deal", added the top minister. "In the event of an incident, you and we shall not remain alone". German Foreign Minister Heiko Maas said that there had at least been progress in moving toward diplomacy.
Further piling pressure on Ankara ahead of an European Union foreign ministers' meeting on Monday and a summit of European Union leaders on Thursday, Cyprus and Greece urged economic sanctions against Turkey over Turkish gas drilling in waters off southern Cyprus.
Friday's remarks were the Pentagon's most explicit criticism of the Turkish operation since it began.
The US still has about 1,000 soldiers in Syria, military officials told CNN and other news outlets.
It was not immediately clear how close the shelling came to the US troops in the mostly-Kurdish town of Kobane, west of the Euphrates River and about 45 miles from the scene of the ground incursion in northeastern Syria, the Newsweek report said, citing a Pentagon official and an Iraqi Kurdish intelligence source. They have been told that U.S. troops will defend themselves if needed, he said. Turkish authorities said two people were killed and three wounded by mortar shelling in the border town of Suruc, raising the death toll to nine civilians killed on the Turkish side.
United Nations Secretary-General Antonio Guterres is "very concerned" by developments in northeast Syria and "any military operation must fully respect the U.N. Charter and worldwide humanitarian law", U.N. spokesman Farhan Haq said.
Darbasiya, some 60km to the east, was also reported to be largely deserted following the exodus.
The Observatory said Turkish forces seized two villages near Ras al Ain and five near Tel Abyad. Kurdish commanders have "given instructions to some of their forces to begin moving north and we're encouraging them not to overreact".
Trump has faced rare criticism from senior figures in his Republican Party who accuse him of deserting USA allies.
The Kurdish forces targeted by Turkey were the US-led coalition's main ground partner in years of battle against Daesh.
Meanwhile, around 30 US House Republicans are expected to introduce a bill to push for sanctions against Turkey.