GENEVA — International relief officials reported an increasingly grim aid crisis stemming from the Syria conflict on Tuesday, with two million people there not getting desperately needed help, and a sudden acceleration of refugees overwhelming the ability of neighboring countries to absorb them.

With less than a week before the start of the Syrian school year, classes have been scrapped indefinitely for tens of thousands of children, because their schools have either been destroyed or been sequestered as squatters’ quarters for displaced families, the officials said. In the province of Homs, so many doctors have fled that only three surgeons remained to serve a population of two million, the officials said, and laws to protect civilians during wartime were being ignored by both government soldiers and insurgents.

The United Nations refugee agency in Geneva said the number of people fleeing Syria had increased almost exponentially, from 18,500 in June to 35,000 in July to 102,000 in August. Roughly 2,000 Syrians were crossing daily into Jordan alone, trying to evade air and artillery attacks on towns near the southern border, said Adrian Edwards, a spokesman for the refugee agency.

The exodus has pushed the number of Syrian refugees to more than a quarter of a million, Mr. Edwards said. Of the total, Jordan now has more than 85,000 refugees and Turkey has more than 78,000, including those who have registered or are awaiting registration. Many more refugees have not registered, and both countries count far greater numbers.