ISTANBUL—President Recep Tayyip Erdogan urged the U.S. and other NATO allies to back Turkey’s military offensive in Syria, threatening to allow waves of Syrian refugees to head for European shores if his country doesn’t receive adequate support.

The offensive, which began Wednesday with air and ground attacks along the border with Syria, targets U.S.-backed Kurdish forces Turkey views as a terrorist threat. It has drawn broad political and popular support in Turkey, but elicited near-unanimous condemnation outside the country—including from the U.S., the EU and China—that has left Ankara isolated.

Mr. Erdogan said 109 Kurdish fighters had been killed, as Turkish troops captured villages on the Syrian side of the border and moved deeper into Syria. The Syrian Observatory for Human Rights put the death toll among Kurdish forces at 23. Six civilians were killed in Turkey on Thursday by mortars fired from the Syrian side, according to Turkish officials, while Kurdish forces reported five civilians had died under Turkish bombs.

A U.S. defense official said that the Turkish offensive consisted largely of ”shaping operations,” strikes and actions to prepare the way for larger-scale ground movements.

The U.S. and other governments have voiced concerns that Turkey’s attack would cause another exodus of civilians from war-ravaged Syria, and distract Kurdish-led forces from combating Islamic State, possibly enabling a resurgence of the extremist group. Kurdish forces have said their priority was now to defend their semiautonomous region.