Turkey's military claims to have seized the key border town of Ras al-Ayn in its attack on the north-eastern border of Syria, a region where many Kurdish people live as well as Arabs, Syrians, and Christians.
The UK says it has 'grave concerns' about the action and has called Turkey to end its military occupation there.
US troops were supporting Kurds in the region until President Trump withdrew them, leaving Kurdish people and Christians there vulnerable to Turkish attack.
Archbishop of Hassaké-Nisibi, Jacques Behnan Hindo says he fears a mass exodus of Christians from the nearby cities of Hassaké and Qamishli.
Describing the plight of 5,000 families in his former diocese, Archbishop Hindo told Catholic charity Aid to the Church in Need (ACN): "In recent days, many had already moved from the border towns to Hassaké."
The archbishop, who reported the killing of two Christians on Thursday, in attacks on Qamishli, added: "Now the conflict has become even more serious and I fear that many will emigrate."
Half of Hassaké Catholics and Orthodox Christians are already thought to have fled with similar numbers exiting Qamishli.
Archbishop Hindo says he "fears a similar exodus, if not a greater one" in light of the escalating conflict.
The archbishop spoke out against the international community's intervention in Syria throughout the countries ongoing conflict and called on them to acknowledge their wrongdoing.
"They acted in Syria for their own interests, hiding behind the ideals of freedom and democracy. Instead they have done nothing but weaken our country at the expense of its own people," he said.
"As always, everyone has their own interests, but it is we Christians who will suffer the consequences."
The United Nations has reported more than 13-thousand Kurds have been displaced by the fighting.