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”.
- What sparked latest violence in Rakhine?
- Annan warns on Rakhine ‘radicalisation’
- Who will help Myanmar’s Rohingya?
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.