“A humanitarian tragedy is unfolding in Sinjar,” Mr. Mladenov said.

In the face of stiff resistance from Shiite militias aligned with Iran that have stalled their march on Baghdad, the ISIS fighters who captured Mosul in June pushed north during the weekend. By Sunday afternoon, they were in control of two other towns after fierce battles with Kurdish security forces, known as the pesh merga, who have been thrust into combat to defend the border of their autonomous region in northern Iraq from encroachments by ISIS.

In a statement, ISIS boasted of conquering “more important areas which were controlled by the pesh merga and the secular militias.” With the new territory, which the group described as “the border triangle of Iraq, Syria and Turkey,” ISIS strengthened its hold on territory that traverses the frontiers of Iraq and Syria, giving it an even greater ability to move fighters and weapons between the front lines of the civil wars in both countries.

According to security officials and residents in the area, the Kurdish forces were routed from Zumar, a town on the road from the Syrian border that also sits on oil fields, and then Sinjar. Sinjar, an isolated city in northwestern Iraq, has been home to a sizable community of Yazidis, Kurdish speakers who ascribe to a religion that combines elements of Islam and ancient Persian religions and who are considered apostates by Muslim extremists.

Later on Sunday, the militants captured Wana, a strategic town near the Tigris River — putting them within striking distance of the Mosul Dam, the country’s largest and an important supplier of electricity and water. The dam is on the Tigris River about 30 miles northwest of Mosul, Iraq’s second-largest city, which fell to ISIS on June 10.

Yazidi residents of Sinjar, who were reached by phone, were terrified. They told of kidnappings and executions of members of their sect. One resident, Sami Hassan, said he was at work at a hospital on Sunday when an injured ISIS fighter arrived and demanded to know the sect to which Mr. Hassan belonged.