Some are reported to have been prevented from crossing into territory controlled by the Syrian government.
"The will of the people can not be vanquished", the civilian administration said in a statement.
Ankara views the YPG as a threat because of its links to the decades-old Kurdish insurgency in Turkey.
The YPG is a key United States ally against IS and played a major role in driving the extremists from much of northern and eastern Syria.
The Turkish ground and air offensive on Afrin raises tensions in the already-complicated Syrian conflict. Although the Turkish state's abuses have been recognised by the worldwide community, the PKK is also broadly treated as a terrorist organisation.
YPG sources told the Kurdish Hawar news agency that its fighters had foiled an attempt to seize the village of Qastal Jando, killing many attackers, and destroyed an armoured vehicle in Qoda.
On Monday, the Kurdish-led SDF announced that they are considering sending reinforcements to Afrin amid the Turkish operation there. Turkey says it intends to create a 30-kilometer (20-mile) deep "secure zone" in Afrin.
Neither side has released much information about fatalities, making the death toll hard to gauge.
Turkish shelling and airstrikes in Afrin have killed 28 civilians, while two civilians were killed as a result of YPG shelling near Azaz, a town held by Turkish-backed opponents of President Bashar al-Assad, the monitoring group said.
Twenty-two of the civilian deaths have been via Turkish air and artillery strikes and two due to Kurdish fire, the Observatory explains, despite Ankara denying that its forces harmed any civilians. In the past few years, Turkey has "so dismantled the rule of law in the country you don't have anymore an independent judiciary or freedom of the press, and civil society has been suppressed".
What is the background to the offensive?
Ankara began "Operation Olive Branch" on Saturday with troops and Syrian rebels moving into north-west Syria's Afrin, an enclave of the Kurdish People's Protection Units (YPG).
US Defence Secretary Jim Mattis said on Tuesday Turkey's offensive was distracting from efforts to defeat Daesh.
Turkish troops and allied Syrian fighters have been trying to take the summit of Bursaya Hill, overlooking the eastern approach to Afrin town.
How have key powers reacted?The U.S. -backed Syrian Democratic Forces (SDF) umbrella group led by the Kurdish YPG said there was no Islamic State presence in Afrin and Turkey had exaggerated the number of dead.
Meanwhile President Recep Tayyip Erdogan has attended the funeral of one of two Turkish soldiers killed in fighting. Both sides agreed that Syria's territorial integrity and sovereignty should be respected.
The U.S. says Russian Federation, a key ally of the government, shares blame for the use of chemical weapons in Syria because it vetoed the renewal of an expert body that determined responsibility for such attacks.
Turkish military forces are utilizing German-manufactured Leopard tanks in the Afrin offensive, resulting in politicians in Berlin condemning the North Atlantic Treaty Organisation ally's use of the vehicles.