"The Afrin operation has de-facto been started on the ground," Erdogan said in a televised speech in the city of Kutahya, without elaborating.
"This will be followed by Manbij," he added, referring to another Kurdish-controlled Syrian town to the east.
Afrin and Manbij are controlled by the People's Protection Units (YPG) Syrian Kurdish militia, which Ankara regards as a terror group.
Turkey has in recent days sent dozens of military vehicles and hundreds of troops to the border area amid repeated threats from top officials that an operation could be launched at any moment.
Turkish forces have over the last two days shelled YPG targets around Afrin and also mobilised pro-Ankara rebel fighters in Syria for the offensive.
"The promises made to us over Manbij were not kept. So nobody can object if we do what is necessary," said Erdogan, referring to past American assurances that the YPG would move out of Afrin.
"Later we will, step by step, clear our country up to the Iraqi border from this terror filth that is trying to besiege our country."
He added that Turkey would "step by step" destroy a "terror corridor" that he said had been set up by the YPG.
Turkey accuses the YPG of being the Syrian offshoot of the Kurdistan Workers' Party (PKK) which has waged a rebellion in the Turkish southeast for more than three decades and is regarded as a terror group by Ankara and its Western allies.
But the YPG has been the key ally of Turkey's fellow NATO member the United States in the fight against Islamic State jihadists, playing a key role in pushing the extremists out of their Syrian strongholds.
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