The refugee problem has particularly unsettled Turkey, which has received at least 78,000 registered Syrians, the most of any country, and has imposed a limit of 100,000. Despite Turkey’s disciplined approach, it has run out of space at its nine camps and is building seven more, but it is unclear what will happen after the limit is reached. Some families have been forced to wait on the Syrian side of the border.

In Lebanon, where a weak central government has been more haphazard in its relief effort than Turkey, the crisis has been aggravated by sectarian fighting that is spilling over from Syria. That has complicated registration efforts by the United Nations refugee agency, which closed its office in Tripoli, in northern Lebanon, on Friday amid days of gunfire.

The plight of the refugees will only grow worse in the coming months, without a significant improvement in accommodations, as the torrid summer gives way to the bone-chilling winter winds that can whip through the unheated tent encampments. And yet, Mr. Edwards said a more immediate problem loomed: Many refugees are housed temporarily in schools, which must be vacated for the fall academic year.

“We are struggling over what to do with people living collectively in Lebanon, in Iraq — groups of refugees who have moved into schools, which have to open up in a matter of weeks,” he said.

The accelerated exodus at least partly reflects increased fighting in Damascus and Aleppo, Syria’s two largest cities, as well as an intense campaign by the government of President Bashar al-Assad to crush insurgents close to the Jordanian border around the southern city of Dara’a, where the uprising began nearly 18 months ago.