Turkey carried out a second day of shelling on Sunday of Kurdish Forces advancing in northern Aleppo province.
Ankara views the Kurdish militia in Syria as allied to the outlawed PKK, which has carried out a decades-long campaign for autonomy in Turkey.
But the United States and others back the Kurdish militia in Syria, the YPG, in its fight against ISIS terrorist group.
Syria has accused Turkey of violating its sovereignty by backing "al-Qaeda-linked terrorists" in the north and has warned it has a right to respond.
"Turkish artillery shelling of Syrian territory constitutes direct support to the armed terrorist organizations," the Syrian government said in letters to the UN Secretary General and the Security Council's Chairman; SANA reported.
The letters accused Turkey of allowing about 100 gunmen - believed to be either "Turkish soldiers or Turkish mercenaries" - to cross into Syria.
"Syria will maintain its legitimate right to respond to the Turkish crimes and attacks and to ask compensation for the damage caused."
France's foreign ministry has also urged Turkey to end its assault on Kurdish fighters in northern Syria.
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