MASNAA, Lebanon—Syrian government forces routed rebels from several Damascus neighborhoods on Friday but rebels struck back in others, as fears of a protracted fight drove some well-to-do supporters of the Assad regime from the capital.

Around 20,000 people have entered Lebanon through this main border crossing since Wednesday, when a rebel bombing struck a meeting of senior regime officials, Lebanese customs officers said. Most of the refugees were families from Damascus, 25 miles to the east.

"The blood that has been spilled in Damascus has scared people," said a native of the capital in her 50s, in Masnaa on Friday. "We do not want to become another Iraq."

As rebels were pounded and driven from the northeastern district of Qaboun on Friday, they began harassing government forces with guerrilla tactics in nearby areas, activists said. "It's a game of alleyways," said an activist based in the city of 2.5 million people.

By nighttime Friday, government forces had begun pounding rebels in another restive district to the southwest, residents said.