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, embattled 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.