BEIRUT, Lebanon — Thousands of exhausted civilians streamed out of the besieged Syrian enclave of eastern Ghouta on Thursday, the largest single-day exodus from the embattled region since the start of a punishing government offensive last month.

The Syrian Observatory for Human Rights, which is based in Britain and tracks the conflict though a network of contacts, said that as many as 20,000 people had fled the region for government-held areas.