The Syrian army and allied forces pounded rebel-held areas of the southwest as thousands of civilians fled to safer opposition held areas along the border with Jordan and Israel, aid workers and rebels said.
They said hundreds of families with their personal belongings had arrived in the last two days in the towns of Tayba and Mataiyah, just a few kilometres from the heavily patrolled border with Jordan.
Busra al Harir, Nahta, Maliha and a string of towns and villages east of Deraa city have borne the brunt of a ramped up assault by the Syrian army begun last week. It is targeting opposition areas in the strategic region bordering Jordan and the Israeli-occupied Golan Heights.
Thousands of Syrians also fled frontline opposition-held villages of Masahra and Hara in the Quneitra province to makeshift camps near the border with Israel, where Syrian artillery avoid shelling, two residents said.
The army's offensive threatens to wreck a de-escalation zone agreed by the United States and Russia last year and risks drawing Washington deeper into the war.
On Saturday, rebels said they had repelled several attempts to seize villages in the Laja region, a volcanic, rugged area that lies between Deraa and Sweida provinces, where most of the ground forces clashes have so far taken place.
"The army is intensifying its campaign and continuing its mobilisation on several fronts," said Abu Bayan, head of Liwa Suquour al Jnoub Free Syrian Army faction.
State media said "terrorists" had fired mortars on Dama and Shomara villages in the Laja region while several rural villages located west of the mainly Druze-inhabited Sweida city were targeted by insurgents.
The United States on Thursday reiterated its demand that the zone be respected, warning Assad and his Russian allies of serious repercussions over violations.
It accused Damascus of initiating air strikes, artillery and rocket attacks.
The southwest is of strategic concern to US-allied Israel, which has this year stepped up attacks on Iran-backed militia allied to Assad.
US ally Jordan which has been worried by the escalation said it was engaged in intensive diplomacy with Washington and Moscow to preserve the zone and prevent a wider confrontation.
The kingdom, already burdened with hundreds of thousands of refugees fleeing war-torn Syria, fear a spillover of refugees along the border if fighting escalates.