A number of US military advisers were present alongside local allied forces, and the Americans led a punishing response that other officials said killed about 100 of the assailants.
Earlier, President Recep Tayyip Erdogan said a helicopter had been shot down without saying who was responsible.
The Turkish army confirmed the crash but did not comment on a cause.
"One of the two Atak helicopters of ours operating on Afrin region unfortunately collapsed".
As Turkish Armed Forces continued to deploy military vehicles at the border close to the operation area, smoke was seen rising from various fields in Syria.
Putin and Erdogan also underscored the importance of "strict and unfailing adherence" to the Astana agreements on de-escalation zones in Syria, the statement said.
The Turkish government, in turn, tried to present its fight against YPG - effectively a branch of long-time enemy number one, the Kurdistan Workers' Party (PKK) - as a battle against the Islamic State (IS).
Eleven Turkish soldiers were killed on Saturday, including two military personnel when a helicopter was downed, in the bloodiest day in Ankara's offensive against a Kurdish militia in northern Syria.
The Turkish military later said nine more soldiers were killed in separate incidents but did not give details.
Since the war began in 2011, Syrian Kurdish fighters and their allies have set up three autonomous cantons in the north, including Afrin which borders Turkey.
A spokesman for the Kurdish militia said his fighters downed the chopper in Raju, northwest Afrin.
But Russia, Assad's most powerful ally, said on Thursday a ceasefire was unrealistic and that militants were to blame for the bloodshed.
The Syrian Observatory for Human Rights, a monitoring group, says at least 68 civilians have been killed.
The military has also said that only terrorist targets are being destroyed and "utmost care" is being shown to avoid harming civilians.
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