Artillery fire and gunshots were reported near several border villages, expanding the area of the fighting that flared again Thursday after a land mine explosion along the border wounded five Thai soldiers.
Cambodian and Thai officials claimed to have acted in retaliation.
On Saturday, US President Donald Trump said he is in contact with the leaders of Thailand and Cambodia and has urged both sides to reach a ceasefire.
In a post on social media, Trump suggested he would not conclude a trade agreement with either country if the cross-border hostilities continued.
Both countries have recalled their ambassadors and Thailand closed its northeastern border crossings with Cambodia.
More than 168,000 people have been displaced in the conflict.
Cambodian authorities on Saturday reported 12 new deaths, bringing its toll to 13, while Thai officials said a soldier was killed, raising the deaths to 20, mostly civilians.
The regional bloc, the Association of Southeast Asian Nations, or ASEAN, is under growing pressure to defuse the situation between its two members.
During an emergency meeting on Friday, members of the UN Security Council called for de-escalation and urged ASEAN to mediate a peaceful solution.
The 800-kilometre frontier between Thailand and Cambodia has been disputed for decades, but past confrontations have been limited and brief.
The current tensions broke out in May when a Cambodian soldier was killed in a confrontation that created a diplomatic rift and roiled Thailand's domestic politics.
Cambodia's Defence Ministry condemned what it said was an expanded Thai offensive early on Saturday after five heavy artillery shells were fired into multiple locations in the province of Pursat, calling the attack an "unprovoked and premeditated act of aggression".
Ministry spokesperson Lieutenant General Maly Socheata said tensions flared in the province of Koh Kong, where four Thai naval vessels were reportedly stationed offshore and four others en route.
She said the naval deployment was an "act of aggression" that risked further escalation.
Maly Socheata said seven civilians and five soldiers were killed in two days of fighting.
Earlier, one man was reported dead after a pagoda he was hiding under was hit by Thai rockets.
The Thai army had denied targeting Cambodian civilian sites and accused Phnom Penh of using "human shields" by positioning their weapons near residential areas.
Thailand's navy accused Cambodian forces of initiating a new attack in the province of Trat, saying Thai forces responded swiftly and "successfully pushed back the Cambodian incursion at three key points", warning that "aggression will not be tolerated".
Thai authorities also alleged several Cambodian artillery shells had landed across the border in Laos, damaging homes and property.
Lao officials have not publicly responded to the claim.
Cambodia's Information Minister Neth Pheaktra said on Saturday the clashes had forced 10,865 Cambodian families, or 37,635 people, in three border provinces to evacuate, while Thai officials said more than 131,000 people had fled their border villages.
Human Rights Watch urged the UN Security Council to press the Thai and Cambodian governments to abide by international humanitarian law and take all steps to protect civilians.
Children have been harmed and Thai authorities have closed at least 852 schools and seven hospitals for safety reasons, the rights group said.
Both sides have employed rocket and artillery attacks.
After initially denying Cambodian claims that internationally prohibited cluster munitions were being used, a Thai military spokesperson said that such weapons could be used: "when necessary" to target military objectives.
HRW condemned the use of cluster munitions in populated areas.
