The Syrian Observatory for Human Rights said the clashes erupted "after the Kurdish People's Protection Units (YPG) fired on an armoured vehicle belonging to Turkish forces that had crossed the Syrian-Turkish border."
Turkish forces were firing artillery at YPG positions west of Darbasiyah, a border town in Syria's northeast Hasakeh province, and YPG fighters were lobbing rockets on Turkish outposts, said the Observatory's Rami Abdel Rahman.
There was no immediate information on casualties.
The YPG on Wednesday warned of "widening aggression," and said Turkey had shelled at least three separate areas along the border, including Darbasiyah.
The cross-border violence came a day after deadly Turkish air raids on a YPG base about 150 kilometres (100 miles) further east.
Tuesday's strikes killed 28 people, most of them YPG members, and left another 19 wounded, according to the Observatory.
Ankara considers the YPG a "terrorist" organisation because of its links to the outlawed Kurdistan Workers' Party (PKK), and vowed on Tuesday to continue acting against such groups.
The US State Department said it was "deeply concerned" that the strikes were conducted "without proper coordination either with the United States or the broader global coalition" that is fighting IS in Syria and neighbouring Iraq.
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