A media centre affiliated to the Kurdish administration also said that Ras al-Ain and Tal Abyad, which have been nearly emptied of their population, were the worst hit.
Reuters reporters heard gunfire there from across the frontier in the Turkish town of Ceylanpinar.
Some made victory signs, shouted "Allahu akbar" (God is great) and waved Syrian rebel flags as they advanced towards Ras al-Ain.
Some 120 km (75 miles) to the west, Turkish howitzers resumed shelling near the Syrian town of Tel Abyad, a witness said.
North Atlantic Treaty Organisation member Turkey considers the Syrian Kurdish fighters to be terrorists linked to a Kurdish insurgency inside Turkey and says the offensive is a counterterrorism operation necessary for its own national security.
Corabatir, who is the president of the Research Center on Asylum and Migration and is a former spokesman in Turkey for the United Nations refugee agency UNHCR, said forced deportations are against global laws and Turkey's open-border policy. Civilians and SDF fighters have been killed, and dozens have been wounded.
On Sunday President Donald Trump appeared to give a green light to a Turkish invasion, announcing that Washington would pull back several dozen United States special forces fighters positioned on the Syria-Turkey frontier.
He said that the so-called "safe zone" in northeastern Syria as envisaged by Turkey is unlikely would satisfy global criteria for refugee return as laid down by UNHCR.
Meanwhile, around 30 US House Republicans are expected to introduce a bill to push for sanctions against Turkey. "That is why I consider yesterday's threats made by President Erdogan totally out of place", Donald Tusk, president of the European Council, said on October 11 after his meeting with President of Cyprus Nicos Anastasiades.
President Recep Tayyip Erdogan's government considers the Kurds "terrorists" who are allied with Turkey's outlawed Kurdistan Workers' Party. In a few places, the Turkish army might go a kilometer past the limit, he added. "Its action also has the potential for causing humanitarian and civilian distress", India said in a statement.
The International Rescue Committee helps organization said that 64 000 people had already reportedly fled their homes.
Trump threatens to 'obliterate' Turkey's economy
Early on Monday, US-backed Kurdish led forces in Syria said US troops had begun withdrawing from areas along Turkey's border. The Turks would become responsible for all Islamic State (IS) group prisoners in the area, the United States statement said.
The YPG is the main fighting element of the Syrian Democratic Forces (SDF), the principal ally of the USA in Syria against the ISIL group.
The U.N. refugee agency said tens of thousands were on the move, and aid agencies warned that almost a half-million people near the border were at risk - in scenes similar to those from a few years ago, when civilians fled the Islamic State group militants.
Hungary, which has refused to take in people fleeing Syria's eight-year civil war, is against sanctions and blocked an European Union statement on Wednesday criticising Turkey, two diplomats said.
At this stage, the countries have not gained the support from all council members for the declaration, diplomats said.
In statements Thursday, Trump said that options for dealing with the situation included sending in "thousands" of American forces, a proposal he rejected; imposing severe economic sanctions on Turkey; or providing United States mediation between Turkey and the Kurds.
USA personnel at the base in northern Syria said they could not speak to CNN due to the sensitive nature of their work here.
Without elaborating, he said the USA was "going to possibly do something very, very tough with respect to sanctions and other financial things" against Turkey.
Western countries' rejection of the Turkish offensive creates a rift within the North Atlantic Treaty Organisation alliance, in which Turkey is the main Muslim member.
He said the Turks "know exactly where American forces are" throughout Syria and in the invasion zone. Cavusoglu said Ankara expected "strong solidarity" from the alliance.
The Turkish invasion also has raised the prospect of losing control of thousands of captured Islamic State fighters. "When you say they're fighting with the USA, yes, but they're fighting for their land".