Nearly 40,000 Rohingya Muslims have fled to neighbouring Bangladesh since violence erupted in Myanmar's Rakhine state a week ago, the UN estimates.
Scores of people are reported to have died since Rohingya militants attacked police posts on 25 August.
Subsequent clashes have sent civilians from all communities fleeing.
Many Rohingya are trying to cross the Naf river to reach Bangladesh. On Friday, 16 more bodies were found washed up on the shore.
Their discovery brings the number of people believed to have died in capsized boats to about 40.
Mainuddin Khan, police chief of the Teknaf border town, told AFP news agency that the group included a young girl, and said they "had been floating in the river for a while".
On Thursday the UN Special Rapporteur on human rights in Myanmar, Yanghee Lee, said the "worsening cycle of violence" in Rakhine was of "grave concern and must be broken urgently".
'All space occupied'
UN officials in Bangladesh say 38,000 people have now crossed the border, while cautioning that the figure is an estimate.
"We are seeing lots of makeshifts tents and shelters on the side of the road - every available space is being occupied," UNHCR regional spokeswoman Vivian Tan told the BBC.
Ms Tan said they had heard tales of people being shot as they tried to cross the border, but that it was not clear who was firing.
Thousands more people are waiting to cross the border, reports say. Ms Tan said border controls appeared to vary from place to place.
Rohingya activists, and those who have fled, say the military and Buddhist mobs have launched a retaliatory crackdown after the police-post attacks that left 12 members of the security forces dead. There are reports of soldiers burning Rohingya villages and attacking residents.
The Burmese military says it is fighting the militants, whose ranks may have been swelled by Rohingya men. It says the militants are behind the fires. There have also been reports of attacks by the militants on members of other communities.
Independently verifying the situation on the ground is difficult because journalists cannot gain unrestricted access. A Reuters report citing the military said about 400 people had died since the attacks last week, but this cannot be confirmed.
Journalists report seeing fires from across the Bangladesh border and satellite images also show several blazes or burned areas.
On Thursday US Ambassador to the UN Nikki Haley said the security forces had "a responsibility to adhere to international humanitarian law, which includes refraining from attacking innocent civilians and humanitarian workers and ensuring assistance reaches those in need".
Rakhine, the poorest region in Myanmar, is home to more than a million Rohingya. They have faced decades of persecution in Buddhist-majority Myanmar, where they are not considered citizens.
There have been waves of deadly violence in recent years. The current upsurge is the most significant since October 2016, when nine policemen died in attacks on border posts.
Until then there had been no indication of an armed insurgency, despite the ethnic tensions.
Both the attacks in October and on 25 August were carried out by a group called the Arakan Rohingya Salvation Army (Arsa).
It says its aim is to protect Muslim Rohingya from state repression in Myanmar. The government says it is a terrorist group.
The military also carried out a crackdown after the attacks in October that led to widespread allegations of rape, murder and torture. Tens of thousands of Rohingya fled to Bangladesh then.
The UN is now carrying out a formal investigation, although the Burmese military denies wrongdoing.