BEIRUT (Reuters) - Syrian government forces pounded rebel-held areas of the southwest with artillery on Thursday, in a steady escalation by President Bashar al-Assad who has vowed to win back the area bordering Jordan and the Israeli-occupied Golan Heights.

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The Syrian Observatory for Human Rights said 12,500 civilians had fled the town of Busra al-Hariri and nearby areas of Deraa province in the last two days. Rebel officials gave conflicting assessments of the scale of displacement.

A major government offensive in the southwest risks further escalating the seven-year-old war. The United States has warned it would take “firm and appropriate measures” in response to government violations of a “de-escalation” deal it brokered with Russia last year to contain the conflict in the southwest.

The conflict has pivoted toward the southwest since Assad, with critical help from his Russian and Iran-backed allies, crushed the last remaining pockets of rebel-held territory near Damascus and the city of Homs.

Although he has vowed to win back the area, a military campaign to recover it is set to be complicated by the interests of both U.S.-allied Jordan and Israel, which is deeply alarmed by Iran’s role in Syria.

The U.S. State Department said it was “deeply troubled by reports of increasing Syrian regime operations” in the area.

“Syrian regime military and militia units, according to our reports, have violated the southwest de-escalation zone and initiated air strikes, artillery, and rocket attacks,” spokeswoman Heather Nauert said in a statement, warning the Russian government and the Assad government of “serious repercussions” of the violations.

While Syrian government forces have deployed artillery in the attacks so far, there have been few air strikes. Air power has been critical in helping Assad defeat rebels in places such as Aleppo and eastern Ghouta.

The Syrian state news agency SANA said the army was carrying out “focused artillery strikes” on militant positions in the towns of al-Harak and Busra al-Harir. State media said rockets fired by rebels from the Deraa area had killed two people in the government-held city of Sweida to the east.

A rebel commander, Colonel Nassim Abu Arra, commander of the Youth of Sunna Forces, said government forces seemed intent on capturing Busra al-Harir and the surrounding area, where he said shelling had escalated to a degree.

Abu Bakr al-Hassan, spokesman for another southern rebel group, Jaish al-Thawra, described the shelling in that area as heavy. “The civilians are afraid that the random bombardment is the start of a campaign the regime and the Iranian militias have been planning to launch in the south,” he told Reuters.

The United Nations was “concerned about reports of an escalation of violence in Deraa governorate”, which was “endangering civilians and displacing hundreds of families”, Jens Laerke, the U.N. humanitarian spokesman in Geneva, said.

Assad said earlier this month the government, at Russia’s suggestion, was seeking to strike a deal in the southwest similar to agreements that have restored its control of other areas through withdrawals of rebel forces.

But he also said there had been no results yet and blamed “Israeli and American interference”.

He said the territory would be recovered by force if necessary. Rebels have vowed not to surrender “an inch” of the territory to Assad, one of their commanders said this week.