IS seized large parts northern Iraq from government control in a major offensive in June.
The UN special envoy to Iraq, Nickolay Mladenov, said that a “humanitarian tragedy is unfolding in Sinjar”.
“The United Nations has grave concerns for the physical safety of these civilians” he said.
“The humanitarian situation of these civilians is reported as dire, and they are in urgent need of basic items including food, water and medicine” he added.
The UN said many of those who fled are in exposed areas in mountains near the town.
Many of those in Sinjar are believed to have fled from earlier IS advances in northern Iraq.
The town is home to a large community of Kurdish Yazidis, whom IS consider heretical.
Two Yazidi shrines have reportedly been destroyed in the town.
Peshmerga retreat
Kurdish military forces, known as the Peshmerga, were also forced to retreat from the nearby town of Zumar on Saturday after a militant offensive.
Kurdish forces had held the town since the Iraqi army retreated from the are in June.
Eyewitnesses said militants also seized control of two small oilfields near Zumar.
IS already controls several other oil installations in northern Iraq, which are believed to fund its activities.
Iraqi state television reported that the militants also took control of Mosul Dam on Sunday after the withdrawal of Kurdish forces.
The dam is the largest in Iraq and provides much of Mosul’s electricity.
Further south, clashes between the Iraqi army and sunni militants continued near the town of Jufr al-Sakhar, military officials said.
The Iraqi army said it conducted several airstrikes on militants in the centre of the town, which lies about 60 km south-west of Baghdad.
The town was captured by the militants last week.
The fighting this summer has been one of the worst crises to hit Iraq since the withdrawal of US forces in 2011.
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