The map shows the situation in Afrin canton in north-west Syria. The Turkish army and associated Takfiri militia (green) are trying to push the Syrian Kurds (yellow) out of the area.
Syrian Afrin Turkey battle situation
© South Front
What is claimed as newly Turkish conquered (light green arrows) on the map may overnight change back into Kurdish hands. This has happened several times over the last week. The fighting is quite intense (vid). The Kurds seem to have many anti-tank guided missiles (ATGM) and the Turks have all the tools of a professional army plus a fanatic militia that serves as their infantry. These (vid) Turkish sponsored Takfiris in the Afrin area seem to come from various countries. One sings a Nasheed:
"...
We were steadfast in Grozny,
We were steadfast in Dagestan.
And we took Tora Bora as our den,
And we took Tora Bora as our den.
We made glory in Afrin and in sad Aleppo.
And Afrin is calling for us: 'Oh welcome good people.' "
Grozny and Dagestan, both in the Russian Federation, were devastated in wars against Takfiris. The Tora Bora caves in east-Afghanistan were al-Qaeda's stronghold before the U.S. waged war on the country.

Here is another video of Erdogan's henchmen:
SURA‏ @AlSuraEnglish - 4:26 PM - 1 Feb 2018
#BREAKING - (Graphic Warning) #Turkish-backed #Syrian rebels strip and mutilate dead female #Kurdish #YPG (YPJ) fighter in #Afrin as offensive against #Afrin continues.
The graphic video, available through the tweet above (or on request through me), shows a dead young woman, partially clothed in military attire, who's breasts have been cut off. Some ten men in mixed military-civil cloth are standing around her in the 13 seconds video. They can not be identified.
SURA @AlSuraEnglish
Laughter is heard before #Syrian rebel pushes her chest with his boot, another says "No don't do things like this guys". The next fighter says "Bring the other one" referring to the female #Kurdish #YPG (YPJ) fighters corpse. #Turkey has not commented on rebel behavior yet.
I am not sure these are "Syrian militia" or "Syrian rebel" as SURA says - see first video above.

These marauding groups are Erdogan's neo-ottoman version of the Bashibazouk:
A bashi-bazouk (Turkish: başıbozuk, lit. "damaged head", roughly "leaderless" or "disorderly") was an irregular soldier of the Ottoman army. Bashi-bazouk could be ethnic Turks or from other peoples of the empire such as Circassians, Arabs, Albanians, or Bosniaks.
...
They were armed and maintained by the government, but did not receive pay and did not wear uniforms or distinctive badges. They were motivated to fight mostly by expectations of plunder.
...
The bashi-bazouk were notorious for being brutal and undisciplined, thus giving the term its second, colloquial meaning of "undisciplined bandit" in many languages.
After the Syrian Army had re-taken the Al Dahur airbase last week it made a short pause. Some secondary units are cleaning up the black cauldron that is held by a few al-Qaeda and ISIS remnants. The plan for the main Syrian army force is to proceed west-north-west towards Idelb. Idleb governorate is held by al-Qaeda and an assortment of other "rebels". Their lines are currently thin as Turkey had pulled many of these out to support its operation in Afrin.
Syria Abu Al-Duhur ISIS pocket
© South Front
Three days ago Turkey sent a battalion of its regular army, protected by al-Qaeda, to set up a blocking position west of the current Syrian army line. It was an unexpected surprise move. The convoy stopped before reaching its planned destination when Syrian artillery and some aerial bombing hit its vicinity. The next day, maneuvering aimlessly in al-Qaeda country, the convoy was hit by an IED which killed one driver and wounded several soldiers. It is likely that some elements of al-Qaeda, or of other Takfiri groups, disliked the Turkish presence. The convoy then pulled back to its starting point in Turkey.

Today the main Syrian army force started to move again and took several villages north and west of its line. Its next targets are Saraqib and Maarat Al-Numan, both about 15 kilometers away from the current positions, and then a clean-up of the whole are east of the M5 highway between Hama and Aleppo.

PS: Vanessa Beeley just published her excellent research on the highly professional propaganda support for al-Qaeda's "White Helmets": WHITE HELMETS: Channel 4, BBC, The Guardian - Architects of War