Terrified families piled onto trucks and other vehicles, clogging muddy rural roads in yet another harrowing exodus in the conflict, now in its ninth year.
The regime's attacks also killed 13 civilians.
Government forces battling rebels in northwestern Syria took full control on Tuesday of a key highway, over which they lost control in 2012, linking the country's four largest cities.
Associated Press video showed the helicopter spiraling from the sky and breaking up as fire poured from its fuselage, just before it crashed.
An AFP correspondent saw the bodies of the two pilots and the mangled remains of the helicopter at the site of the crash near the village of Qaminas, southeast of Idlib city.
Amid the fighting near Nairab, insurgents shot down a government helicopter gunship, killing the two crew members onboard, according to the Britain-based Syrian Observatory for Human Rights and Step news agency, an activist collective.
The temperature in some parts of Idlib has now dipped to minus 11 degrees Celsius, making living conditions in improvised, unfinished structures even more unbearable, he said.
Syrian refugees head northwest through the town of Hazano in Idlib Province as they flee renewed fighting, January 27, 2020.
Last week's attack by regime forces in Idlib killed seven Turkish soldiers and one civilian contractor working with the Turkish military and injured more than a dozen people.
Syrians drive through the city of al-Mastouma in Idlib province as they flee a government offensive, January 28, 2020.
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In a speech Tuesday in Ankara, Turkish President Recep Tayyip Erdogan warned the Syrian government that it will "pay a very, very heavy price" for attacks on Turkish troops.
"The Syrians want to get rid of the radical groups in Idlib, and Turkey is using these groups as an instrument for the presence of Turkey in northern Syria", said global relations professor Huseyin Bagci of Ankara's Middle East Technical University.
On Sunday raids by regime ally Russian Federation left 14 people dead, including nine in the village of Kar Nuran in southwestern Aleppo province, the monitor said.
Turkey has backed some anti-regime groups in the eight-year conflict and set up military posts in Idlib under a 2018 deal with Russian Federation.
"We have given the necessary response to the Syrian side at the highest level", Erdogan said.
Cavusoglu added that Erdogan and Russian President Vladimir Putin would talk by telephone. "But it doesn't mean we won't talk to them", a Turkish official source told MEE, speaking on condition of anonymity.
It's not the first shelling incident between Turkish and Syrian forces in Idlib.
The Observatory also reported that Syrian troops have secured the highway that links southern Syria with the northern city of Aleppo.
Turkey announced last week that it had sent "major reinforcements" to the region to stem government attacks, adding: 'All options are on the table'.
The highway, known as M5, starts in the country's south near the border with Jordan and moves north all the way to Aleppo.