Thailand’s leader has warned that clashes between Thailand and Cambodia, which have killed at least 16 people and displaced tens of thousands in both countries, could “move towards war”.

Acting Prime Minister Phumtham Wechayachai added that the fighting now included heavy weapons and had spread to 12 locations along the border.

Thailand also accused Cambodia of firing into civilian areas and evacuated all villages deemed to be within the radius of its rockets.

Cambodia, for its part, accused Thailand of using cluster munitions. Cluster munitions are banned in much of the world because of their indiscriminate effect on civilian populations. Thailand has not responded to the allegations.

Meanwhile, Thailand’s foreign minister told the Reuters news agency that there was “no need” for third-party mediation in the conflict, even as global leaders appealed for an immediate ceasefire.

Malaysian Prime Minister Anwar Ibrahim, who chairs the Association of South East Asian Nations, or Asean, had earlier offered to facilitate talks between the two countries.

“I welcome the positive signals and willingness shown by both Bangkok and Phnom Penh to consider this path forward,” Anwar wrote on Facebook late Thursday.

But Thai foreign ministry spokesman Nikorndej Balankura told Reuters that the situation must be solved through bilateral means, adding that Phnom Penh must stop its offensive moves.

“I don’t think we need any mediation from a third country yet,” Nikorndej said.

The comments came as both countries updated the death toll from the fighting.

Thailand said 14 civilians and one soldier were killed in the fighting, while provincial authorities in Cambodia said at least one civilian in Oddar Meanchey was killed.

The US has also called for an “immediate cessation of hostilities, protection of civilians and a peaceful resolution of the conflict”.

“We are… gravely concerned by the escalating violence along the Thailand Cambodia border, and deeply saddened by reports of harm to civilians,” State Department spokesman Tommy Pigott said at a regular media briefing.

China, which has political and strategic ties with Cambodia and Thailand, said it is “deeply concerned” over the conflict and hopes both sides can resolve issues through dialogue and consultation.

Australia, the European Union and France have also called for peace.

The United Nations Security Council is expected to meet on Friday over the conflict.

In a letter to the council on Thursday, Cambodia’s Prime Minister Hun Manet had urged it to intervene to “stop Thailand’s aggression”.

Thailand and Cambodia have each accused the other of firing the first shots on Thursday.

Thailand claims the clash began with Cambodia’s military deploying drones to conduct surveillance of Thai troops near the border.

Cambodia says Thai soldiers initiated the conflict when they violated a prior agreement by advancing on a Khmer-Hindu temple near the border.

The dispute between the two countries dates back to more than a hundred years ago, when the borders of the two nations were drawn after the French occupation of Cambodia.

There were sporadic clashes over the years which saw soldiers and civilians killed on both sides.

The latest tensions ramped up in May after a Cambodian soldier was killed in a clash. This plunged bilateral ties to their lowest point in more than a decade.

At a sports complex that was converted into an evacuation centre in Thailand’s Surin province, evacuees – many of them children and elderly – said they were still shaken by the rocket and artillery attacks they witnessed on Thursday.

Older evacuees who had lived through bombardments during the Cambodian Civil War of the 1980s told the BBC the recent fighting was the worst they had experienced.

Joi Phasuwan, from the Phanom Dong Rak district, said she and her two grandchildren “waited a long time” before they finally got moved to the evacuation centre. They moved farther this time compared to past evacuations, she added.

Elsewhere, a small group of older men huddled around the entrance to a school about 10km (6.2 miles) from the border.

All the houses around them were deserted. The men stayed back to guard their homes and livestock, while all the women and children had been evacuated.

They stayed close to an improvised shelter, constructed last month from sections of concrete pipe and reinforced with sandbags and rows of sand-filled tires.

Artillery boomed across this small village all morning, and the Cambodian rocket attacks on Thursday unnerved the men.

The school was also being used by Thai soldiers, but they did not want to be filmed – nor did they want the school to be identified.

The roads were empty, apart from the occasional military lorry ferrying soldiers. Passing through village after village, there was no-one to be seen.

Life all along this border region has been brought to a halt by a dispute over a few old temples and some insignificant strips of forested land.

Additional reporting by Lulu Luo in Surin