"Fighting is ongoing inside the city, where Turkish forces and allied rebels have seized some neighborhoods," the Britain-based Syrian Observatory for Human Rights monitoring group said.
Pro-Ankara rebels said they "broke into the city from the eastern and western sides" to seize the neighborhoods of Ashrafieh and Jamiliyyeh.
Civilians hiding in basements inside the city could hear fighting outside and people shouting "God is greatest", one resident told AFP.
Turkish-led forces have advanced rapidly into the Kurdish enclave around Afrin city near the Turkish border since launching an assault on it almost two months ago.
They are fighting the Kurdish People's Protection Units (YPG), which Ankara considers an offshoot of the outlawed Kurdistan Workers' Party (PKK).
The Observatory says more than 280 civilians have been killed in the offensive since January 20, but Ankara denies the reports and says it takes the "utmost care" to avoid civilian casualties.
More than 200,000 civilians have fled Afrin city in the past three days, the Observatory says.
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