BEIRUT — Rebels pushed forward in their battle with the Syrian Army in Damascus on Friday, clashing with regime soldiers in contested neighborhoods in the northeast and shutting down a key highway out of the capital with a row of burning tires, activists said.

In Geneva, the UN refugee agency said there has been a huge increase in the number of people fleeing Syria, with 5,000 refugees crossing the borders daily into neighboring countries. Agency spokesman Adrian Edwards said Friday the mass exodus ‘‘is really a full-on crisis right at the moment.’’