BAGHDAD—An aid agency says more than 5,000 Syrian refugees fleeing a bloody government crackdown have now crossed into northern Iraq and their numbers are growing. Iraq's Kurdish autonomous region is seeking help in coping with them.

The International Organization for Migration said Tuesday that most refugees are ethnic Kurdish Sunni Muslims, some paying around $300 to be smuggled into Iraq by car or by foot.

It said a refugee camp has grown to 3,500 in the past month and local officials have sought help with sanitation and supplies. About 800 more are staying in mosques.

Activists say more than 13,000 have been killed in Syria since the uprising began in March 2011. Europe's humanitarian aid commissioner has said there are 95,000 refugees, mostly in Turkey, Lebanon and Jordan.