While Turkish Foreign Minister Mevlut Cavusoglu says Turkey has notified all parties involved - including the Syrian government - about the offensive, Damascus denied this was the case.
Turkish artillery and war planes pounded YPG sites around Afrin and total of 153 targets, including YPG refuges and weapons stores have now been hit, according to the army.
The Syrian-Kurdish YPG militia, supported by the United States but seen as a terrorist organisation by Turkey, said it had repulsed the Turkish forces and their allies after fierce clashes.
The U.S. created some Kurdish populated areas in Syria and tried to isolate these regions, according to the statement. "God willing, we will complete this operation very quickly", said the Turkish president.
Turkey's incursion is a direct challenge to Washington, which has openly supported the Kurdish militia against ISIS and which has made the YPG a critical part of its plans for a long-term USA presence in Syria.
The Kurdish group targeted, known as the YPG, is active in the Afrin region, across from Turkey's southern border. The United States is backing the YPG in Syria, seeing it as an effective partner in the fight against Islamic State.
Turkey could also face blowback from the Kurdish insurgency within its own borders.
In retaliation, Kurdish forces fired several rockets into 2 Turkish border towns, killing one person and wounding about 30 others.
"Syria calls on the global community to condemn this Turkish aggression and immediately take the steps necessary to stop it", a representative from the Syrian Ministry of Foreign Affairs told SANA.
Some 25,000 FSA fighters have joined the offensive on the Turkish side, rebel commander Maj Yasser Abdul Rahim told Reuters.
A green light from Russian Federation, which has a military presence in northern Syria, is seen as crucial in ensuring the operation is a success. Russia, too, has called for restraint.
Russia's defense ministry said the main trigger for the escalation was provocative steps taken by the USA aimed at isolating areas with predominantly Kurdish populations.
Ankara fears the establishment of a Kurdish corridor along its border.
But Lavrov on Monday said that Syrian Kurds should play a role in the "future political process".
The United States, Turkey's North Atlantic Treaty Organisation ally, has warned against the operation against the YPG to whom it is lending military support in the fight against IS, a move that has fiercely angered Ankara.