The Syrian Observatory for Human Rights said at least two of the civilian victims were killed in artillery strikes on the city of Qamishli.
Turkey launched its threatened offensive hours earlier, with a limited number of air strikes and mostly artillery fire across most of the width of its long border with Kurdish-controlled regions of Syria.
The Britain-based Observatory, which has a vast network of sources across Syria, said more than 40 people had also been wounded in the early phase of the attack.
The Kurds, whose main militia is considered a terrorist organisation by Ankara, have vowed to resist any ground offensive and called on the population to mobilise.
The flat and open terrain favours Turkey's vastly superior military however and the Kurdish forces have limited means to resist without the support of their US allies.
The offensive, which was widely condemned, comes three days after US President Donald Trump announced a US troop pullback from the border, effectively green-lighting a Turkish invasion.
Follow us on Twitter: @The_NewArab