BEIRUT — As many as 11,000 people fled Syria in 24 hours, some of them desperately clambering through a razor-wire fence into Turkey on Friday to escape fierce fighting between rebels and government forces for control of a border town.

The exodus is a sign of the escalating ferocity of the violence, which has killed more than 36,000 people since March 2011. Despite the bloodshed, President Bashar Assad insisted there was no civil war in Syria, saying in a rare TV appearance that he was protecting Syrians against “terrorism” supported from abroad.

The flood of Syrians into neighboring Turkey, Jordan and Lebanon was “the highest that we have had in quite some time,” said Panos Moumtzis, the U.N. refugee agency’s regional coordinator for the region.

About 2,000 to 3,000 people are fleeing Syria daily, and the recent surge brings the number registered with the agency to more than 408,000, he said.

The true total of those fleeing the conflict is much higher because many refugees have not registered, Moumtzis said.

During the 24-hour period that began Thursday, 9,000 Syrians crossed into Turkey — including 70 who were wounded and two who then died, U.N. officials said. Jordan and Lebanon each absorbed another 1,000 refugees.

The largest flow into Turkey came from the fighting at Ras al-Ayn in the predominantly Kurdish oil-producing northeastern province of al-Hasaka. The town hugs the border, practically adjacent to the Turkish town of Ceylanpinar.

On Thursday, rebels captured a border crossing between the two towns, Ceylanpinar’s mayor, Ismail Aslan, said by telephone.

Rebels on Friday overran three security compounds in the town belonging to the military intelligence, air force intelligence and general intelligence directorate agencies, according to the Britain-based Syrian Observatory for Human Rights, an opposition activist group.

More than 20 soldiers were killed in the fighting, the Observatory said.

Regime forces shelled rebel positions Friday morning, Aslan said. Regime tanks were also moving in to join the fight, according to another opposition activist group, the Local Coordination Committees.

Syria’s more than 2 million Kurds, long marginalized, have largely stayed out of the fighting although some have taken part in demonstrations against Assad. But like other minority groups, they have increasingly been drawn into the fighting.

The rebel push on Ras al-Ayn, an ethnically mixed town inhabited by Kurds, Arab Muslims and Christians, was likely to inflame tensions with the Kurds, who fear a government offensive to flush out the fighters.

Video from Turkey’s Anadolu news agency showed Syrians jumping over and climbing through the razorwire fence on the 566-mile border to cross into Ceylanpinar.

Others fled into Turkey farther west along the border, trying to escape fighting at the Syrian town of Harem in Idlib province, which has seen intense battles in recent days.

The new arrivals bring the number of refugees in Turkey to about 120,000.

Radhouane Nouicer, the U.N.’s regional humanitarian coordinator for Syria, said the country is seeing unrelenting increases in violence, suffering, displacement and loss, “and civilian Syrians continue to pay the price.”

He said U.N. officials also worry that Kurds and Palestinians are increasingly being drawn into the fighting.

The Anadolu agency reported a group of Syrian soldiers, including two generals and 11 colonels, had fled to Turkey with their families and were taken to a camp for military defectors, including dozens of other generals.

The surge in refugees fleeing the conflict came as agencies of the United Nations and other groups met donor governments in Geneva to report on the crisis and appealed for a greater financial support.

The United Nations’ estimates that more than 2.5 million people are in need of humanitarian assistance across Syria — including 1.2 million displaced by the conflict — were conservative, Nouicer said.

The United States will provide $34 million in additional aid to Syrians affected by conflict, bringing the total provided by the United States to $165 million, the U.S. diplomatic mission in Geneva said in an announcement distributed at the donor meeting.

In the interview by broadcaster Russia Today and aired Friday, Assad struck a defiant tone.

“We do not have a civil war,” he said, speaking in English. “It is about terrorism and the support coming from abroad to terrorists to destabilize Syria. This is our war.”

He called it a case of “terrorism through proxies, either Syrians living in Syria or foreign fighters coming from abroad.”

Asked if he has any regrets, he said, “Not now,” although he acknowledged that “when everything is clear,” it would be normal to find some mistakes.

Assad, 47, insists that there has been no popular uprising in his country and said he will not step down, hinting he will stay in his post until at least 2014, when elections are scheduled.

“I think for the president to stay or leave is a popular issue,” he said.

He said that when foreign countries stop sending arms to rebels, “I can tell [you] that in weeks we can finish everything.”

The conflict began largely with peaceful protests of Assad’s rule but turned bloody after rebels took up arms in response to the regime’s crackdown. Rebels have driven regime forces out of much of a pocket of northwestern Syria and battle troops in several cities and towns, even as the fight takes on dangerous sectarian tones between a mainly Sunni opposition and a regime dominated by Assad’s minority Allawite sect.

Assad, who came to power after his father and predecessor Hafez Assad died in 2000, said in a part of the interview released Thursday that he will “live and die” in Syria and will not leave his country.

Sophie Shevardnadze, the Russia Today correspondent who conducted the interview at a presidential palace in Damascus, said Assad told her before the session that his British-born wife, Asma, and his three children are still in Syria.

The Observatory said at least 120 people were killed in violence across the country Friday, including 18 who died in intense shelling of the eastern town of Qouriyeh in the Deir el-Zour province, which borders Iraq.

Amateur video posted by Syrian activists showed graphic footage of men, women and children, some of them with gaping wounds, at what appeared to be a market.

Activist videos could not be independently verified because of reporting restrictions in Syria, but they appeared genuine and corresponded to other reporting of the events depicted.

A car bomb near the mayor’s office in the Damascus suburb of Maadamiyeh killed at least four people, the Observatory said.

Syria’s main opposition bloc in exile, the Syrian National Council, chose a Paris-based former geography teacher Friday as its new president. George Sabra, a Christian who once wrote for the Arabic version of Sesame Street, said his election is a sign the opposition is not plagued by sectarian divisions.

But the bloc suffered a major blow when the Local Coordination Committees announced it was withdrawing from the council. In a statement Friday, it said the Syrian National Council has failed to change and was no longer considered fit to be the political representative of the Syrian people.

Adib Shishakly, one of the council’s founders and the grandson of a former president of Syria, also announced his resignation because of the group’s lack of transparency and failure to change.

The council has been widely criticized by U.S. officials and other Syrian opposition groups as being petty, ineffective and cut off from events in Syria.

The resignations came as the council was debating in Doha, Qatar, whether to become part of a single leadership group that would set up a transitional government in rebel-held areas of Syria. Several senior council members said the group is likely to accept the U.S.-backed plan in principle, possibly by the end of Friday, but has significant reservations.

Proponents say the plan could give new momentum to the battle to oust Assad.

Information for this article was contributed by Zeina Karam, John Heilprin, Suzan Fraser, Karin Laub and Bassem Mroue of The Associated Press and by Nick Cumming-Bruce, Sebnem Arsu, Hania Mourtada and Christine Hauser of The New York Times.

Front Section, Pages 1 on 11/10/2012