The Turkish military has said Thursday's air strike, which killed three of its soldiers, was thought to have been carried out by the Syrian air force.
One Turkish soldier has been killed and five others sustained light injuries in clashes with Islamic State (IS) extremist group in northern Syria, the Turkish army has said in a statement.
The incident, which occurred after Russia violated Turkey's sovereignty multiple times, severed ties between Russia and Turkey for months before Russian President Vladimir Putin sent a supportive note to Erdogan in July, when Erdogan successfully quashed an attempted military coup.
The moves are part of Operation Euphrates Shield, which began in late August to rid Syria's northern border area of terrorists.
The operation has been widely criticized both by the Syrian Kurds and Damascus, who have accused Ankara of violating Syria's territorial integrity.
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The city is important to the Syrian regime, which has opposed Turkish involvement in its civil war. Turkey's major military confrontations in Syria have come as a result of challenging the Kurdish forces, though Assad and Turkish strongman Recep Tayyip Erdogan have long maintained an acrimonious relationship. Prime Minister Binali Yildirim said Ankara would retaliate.
Ankara and Moscow repaired ties in October after almost a year of tension, but differences remain over Syria.
Reports said FSA fighters, backed by Turkish tanks and air strikes, continued attacks to gain control of Al-Bab on Friday.
Turkish planes also carried out air strikes against seven ISIL targets in northern Syria, the army said in a statement carried by the state-run Anadolu news agency.
After it takes Al-Bab, Turkish military says, the objective is to clear the whole area and retake the town of Manbij from the SDF, the largely Kurdish militia.