"More than 30,000 people were displaced since yesterday (Wednesday) as a result of Turkish bombardment against the city," the Britain-based Syrian Observatory for Human Rights said.
"They went to areas controlled by the Syrian regime and to the outskirts of the city" under the control of Kurdish militia, the monitoring group said.
It claims to have now almost completely encircled the region's main city of Afrin.
An AFP correspondent inside the city saw hundreds of families cramming into the back of pick-up trucks and onto tractor-drawn carts as they prepared to leave the city via the only remaining exit.
Those who stayed behind formed long queues outside shops to buy bread and various food goods in preparation for a fully fledged siege by Turkish and allied forces.
Afrin is one of the cantons in the self-proclaimed Kurdish autonomous administration in northern Syria and Ankara is worried by the consolidation along its border of a de facto statelet ruled by a group it considers a terrorist organisation.