About 73,000 people had crossed the border since violence erupted on August 25, United Nations High Commissioner for Refugees spokeswoman Vivian Tan said.
Refugees reaching Bangladesh described bombs exploding and Rohingya Muslims being burned alive in a military crackdown.
Another aid official said that more than 50 refugees had arrived with bullet injuries and had been taken to hospitals in Cox’s Bazar.
Both Burma’s security officials and Rohingya insurgents have accused each other of atrocities in Rakhine state.
The military has said that nearly 400 people, most of them insurgents, have died in clashes following rebel attacks on security posts.
Many refugees required immediate medical attention as they were suffering from respiratory diseases, infection and malnutrition.
The medical facilities in the border area were insufficient to cope with the influx and more aid and paramedics were needed, aid workers said.
“We fled to Bangladesh to save our lives,” said a man who only gave his first name, Karim.
“The military and extremist Rakhine are burning us, burning us, killing us, setting our village on fire.”
He said he paid for his family to be smuggled on a wooden boat to Bangladesh after soldiers killed 110 Rohingya in their village of Kunnapara, near the coastal town of Maungdaw.
“The military destroyed everything,” he said. “After killing some Rohingya, the military burned their houses and shops.
“We have a baby who is only eight days old, and an old woman who is 105.”
Satellite imagery analysed by Human Rights Watch shows hundreds of buildings had been destroyed in at least 17 sites across Rakhine since August 25, including some 700 structures that appeared to have been burned down in just the village of Chein Khar Li, the rights watchdog said.
The government claims the insurgents burned their own homes and killed Buddhists in Rakhine.
Long-standing tension between Rohingya Muslims and Buddhists erupted in rioting in 2012, forcing more than 100,000 Rohingya into displacement camps.
Bangladeshi police said on Thursday that three boats carrying refugees had capsized in the Naf River, killing at least 26.