The increasingly harsh global condemnation comes as the number of Rohingya who have fled Myanmar's Rakhine state for Bangladesh to escape ethnic unrest hit 389,000, and the United Nations warned of a looming "worst case scenario" with all of the Muslim minority group trying to leave.
The number of refugees was up 10,000 in just 24 hours, as the three-week old crisis deepens.
"We need to support Aung San Suu Kyi and her leadership but also be very clear and unequivocal to the military power sharing in that government that this is unacceptable," Tillerson said Thursday of Myanmar's first civilian leader in decades.
"This violence must stop. This persecution must stop. It has been characterised by many as ethnic cleansing. That must stop," he said during a visit to London, speaking alongside British counterpart Boris Johnson.
Johnson also called on Myanmar's de facto leader to use her "moral capital" to highlight the plight of the Rohingyas.
Suu Kyi, a Nobel laureate and long-time human rights champion, has been condemned for a lack of moral leadership and compassion in resolving the crisis.
She has no control over the powerful military, which ran the country for 50 years.
UN chief Antonio Guterres on Wednesday said the mass displacement of Rohingya amounted to ethnic cleansing.
Amnesty International released fresh satellite images Friday of burned villages in Rakhine state, alleging Myanmar's security forces have led "systematic" clearances of Rohingya Muslim settlements over the last three weeks.
At least 26 villages had been hit by arson attacks in the Rohingya-majority region, the rights group said, with patches of grey ash picked up in photos marking the spots where homes had once stood.
Backing up the pictures, Amnesty said fire sensors also deployed on satellites had detected 80 large-scale blazes across northern Rakhine state since August 25, when the army launched "clearance operations".
"Rakhine state is on fire," said Olof Blomqvist, a researcher with Amnesty International, in a "clear campaign of ethnic cleansing by the Myanmar security forces".
The group quoted Rohingya witnesses who described security officers and vigilantes using petrol or shoulder-fired rocket launchers to set homes alight, before firing on villagers as they fled.
"It's very difficult to conclude that it is anything other than a deliberate effort by the Myanmar military to drive Rohingya out of their own country by any means necessary," Blomqvist added.
'Worst case scenario'
Relief workers are struggling to contain the humanitarian disaster unfolding around the Bangladesh border town of Cox's Bazar with 10,000-20,000 people crossing over each day -- far more than the UN and other agencies had expected.
"We have to estimate the worst case scenario" where all Rohingya flee Rakhine, said Mohammed Abdiker Mohamud, a director of the International Organization for Migration (IOM), the UN's migration agency.
"We cannot just put our heads in sand (and) say that everything will be OK," he added.
"Unless a political solution is found there is a possibility that the entire Rohingya community may come to Bangladesh."
There were previously an estimated 1.1 million Rohingya in Rakhine state, who have endured decades of persecution in Buddhist-dominated Myanmar.
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At least 300,000 had fled to Bangladesh before the latest crackdown started on August 25, following attacks by Rohingya militants on police targets.
The exodus since has taken the overall figure of those who have quit Myanmar to at least 700,000.
Even before arriving to safety in Bangladesh, refugees who have trekked through jungles for days to reach the border are being targeted by profiteering boat operators who have hiked prices 200 times to cross the river separating Myanmar and Bangladesh.
An AFP correspondent at the Naf river said boat owners were charging refugees up to $100 for a 10-30 minute trip that would normally cost less than 50 cents.
"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."
Bangladeshi magistrates operating mobile courts in Cox's Bazar and nearby districts have now started sentencing boat owners and local villagers to terms of up to six months in prison, officials said Thursday.
Once in Bangladesh, refugees - with UNICEF saying 60 percent of new arrivals are children - are faced with desperate conditions in already overstretched camps around Cox's Bazar. UN agencies have warned the country is struggling to cope.
"There are acute shortages of everything, most critically shelter, food and clean water," UNICEF's representative in Bangladesh Edouard Beigbeder said in a statement.
"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."
