Turkey launched operation "Olive Branch" on January 20 against the Kurdish People's Protection Units (YPG), a powerful militia that controls the Afrin region in northwest Syria and which Ankara regards as a "terror group".
On Saturday, the Britain-based Syrian Observatory for Human Rights monitoring group said the attacking forces "are now four kilometres (2.5 miles) away from the town of Afrin from the northeast, where there are intense clashes, air strikes, and artillery fire."
Rami Abel Rahman, head of the Britain-based Observatory, said the aim was to besiege Afrin, cutting it off from other areas.
Ankara says the YPG is the Syrian branch of the Kurdistan Workers' Party (PKK), which has fought a three-decade insurgency in southeast Turkey.
"Olive Branch" initially made slow progress, but the capture on Thursday of Jandairis, a major town in the Kurdish-held enclave, has given the operation a clear shot at the central town of Afrin.
Abdel Rahman said that the advance sparked "major concerns" for tens of thousands of civilians in the town of Afrin.
Thousands of people have fled heavy shelling on the villages and towns near the Turkish border, many of them into Afrin town.
Only one route currently leads out of the town and into government-controlled zones in Syria's northern province of Aleppo.
According to the Observatory, Turkey and allied Syrian opposition factions now hold 60 percent of the Afrin region.
The monitor says at least 370 YPG fighters and 340 rebel fighters have died since the assault began.
It says more than 200 civilians have also been killed, but Turkey denies the reports and says it takes the "utmost care" to avoid civilian casualties.
Ankara says 42 Turkish soldiers have died in the fight.