The town in the northeastern province of Hasaka is controlled by the Kurdish YPG militia, which has been battling Islamic State in Iraq and the Levant (ISIL) terrorist group.
Kurdish fighters have advanced in the last few weeks against ISIL in Hasaka province, notably with the takeover of the town of al Houl by a militant alliance that includes the YPG.
The three blasts, carried out by at least two suicide bombers, hit outside a hospital, at a market and in a residential area in the town of Tel Tamer late on Thursday, the YPG's Redur Xelil said via internet messaging service.
"There is massive destruction in the town, and the number killed is between 50 and 60, all of them civilians," he said, Reuters reported.
The Britain-based Syrian Observatory for Human Rights gave an earlier death toll of at least 22 people in the attack, saying that number was likely to increase.
Amaq News Agency, which supports Islamic State in Iraq and the Levant militants, said in a statement that the group had carried out the attack, targeting Kurdish "bases" in Tel Tamer with three suicide bombs.
The Observatory said one of the bombs exploded near a health centre and another near a vegetable market. It said there was "confirmed information" about casualties among the Kurdish internal security force known as the Asayish.