Meanwhile, Syrian forces have taken more than 80 percent of Eastern Ghouta, and a spokesperson from a main rebel faction told The Daily Star they were in "serious" negotiations with the United Nations about a cease-fire agreement.
Turkish-backed Syrian rebel fighters say they entered before dawn on Sunday from three fronts, meeting no resistance.
For the past two months, Turkey and its own favored Syrian rebels clashed with Kurdish rebels in the Syrian-Kurdish People's Protection Units (YPG).
The capture of Afrin could propel Erdogan to expand the offensive to push the Syrian Kurdish YPG militia from the border and quash aspirations of Kurdish self-rule.
"Most of the terrorists have already fled with tails between their legs", Turkish President Recep Tayyip Erdogan said during a ceremony marking the 103rd anniversary of World War I's Battle of Gallipoli, a Turkish victory laden with symbolic and political weight.
Kurdish civilians are fleeing the city en masse, as Turkish forces tore down the statue depicting a legendary Kurdish figure.More news: No. 13 Marshall knocks off No. 4 Wichita State
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After chasing Kurdish fighters from Afrin, the pro-Ankara fighters broke into shops, restaurants and houses and left with foodstuff, electronic equipment, blankets and other goods, the correspondents said.
Turkey's armed forces say in a statement, troops were combing the streets for mines and improvised explosive devices.
"This is a gift for those who were martyred on March 18", said a Turkish soldier as he waved the national flag in the video. "Maybe it will be cleared by the end of the day - it's empty of (YPG) fighters, they cleared out".
Turkish government spokesperson and Deputy Prime Minister Bekir Bozdağ said the military campaign would continue to secure areas around Afrin and make sure food and medicine were available. "But the project of building a terrorism corridor and building a terrorist state is over", he said. Ankara said the demand did not apply to Afrin, but its operation has faced criticism in the West.
France's foreign minister said Turkey's concerns for its border security did not justify "the deep incursion of Turkish troops in the Afrin zone", which could also weaken global action against remaining Islamic State fighters in Syria.
YPG fighters also escaped before the Turkish-led force moved in but not before burning infrastructure and mining the city.