Some 389,000 Rohingya - including 10,000 in the past 24 hours -- have fled across the border since late August and there have been growing appeals for Myanmar's leader Aung San Suu Kyi to speak out in their defence.
Many have trekked across hills and through jungles for days to reach the border only to be faced with hugely inflated prices for a seat on a boat crossing the Naf river that divides the two countries.
Government spokesman Khaled Mahmud from the Cox's Bazar district told AFP that the mobile courts have convicted 165 people -- of whom 160 people were jailed for between three to six months and the remaining five were fined.
An AFP correspondent at the river said boat owners were charging refugees up to USD 100 for a 10-30 minute trip that would normally cost less than 50 cents -- a mark-up of 200 times.
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She sat with her children beside a highway running along the beach at Teknaf, unable to get a ride to the refugee camps some 10 kilometres (six miles) from the border.
"The boatmen threatened to throw us into the sea if we refused to give them our valuables," said Nadera Banu, 19, who got married only last year but is already a widow.
"I gave up the final memento of my husband, a gold locket given on my wedding day, to escape," she said.
Rohingya living in established refugee camps in Bangladesh have also been accused of joining the profiteering.
The influx has left Bangladesh struggling to provide relief for exhausted and hungry refugees -- some 60 percent of whom are children.
The UN said today it had been caught by surprise by the crisis, with other international agencies and the Bangladesh government also overwhelmed.
Mohammed Abdiker Mohamud, a director of the International Organization for Migration, the UN's migration agency, said it was preparing for the worst.
"We cannot just put our heads in sand say that everything will be OK."
Meanwhile UNICEF said children's health was at risk as conditions deteriorate.
"Conditions on the ground place children at risk of high risk of water-borne disease. We have a monumental task ahead of us to protect these extremely vulnerable children," Edouard Beigbeder, UNICEF's representative in Bangladesh, said in a statement.
The crisis, which began when Myanmar launched a military crackdown on August 25, has sparked international alarm, with the UN Security Council on Wednesday breaking its weeks-long silence and calling for an end to the violence.
The 15-member council expressed concern about the security operations in Rakhine state and called for "immediate steps" to end the violence.
The 1.1-million-strong Rohingya have suffered years of discrimination in Myanmar, where they are denied citizenship even though many have longstanding roots in the country.
Suu Kyi's spokesman has said she will not attend next week's annual meeting of world leaders at the United Nations, where the plight of the Rohingya will be in the spotlight.
He said the Nobel laureate and long-time human rights champion, who has been condemned for a lack of moral leadership and compassion in resolving the crisis, will deliver an address next week on peace and reconciliation in Myanmar.
There is also scant sympathy among Myanmar's Buddhist majority for the Rohingya, who are commonly branded "Bengalis" -- shorthand for illegal immigrants.
Rohingya arriving in Bangladesh have told chilling accounts of soldiers firing on civilians and razing entire villages in the north of Rakhine with the help of Buddhist mobs.
The army denies the allegations.
Twelve Nobel laureates have signed an open letter urging the UN to "intervene immediately by using all available means" to end the "crimes against humanity" unfolding in Rakhine.