The figures from the International Organization for Migration (IOM) on Wednesday came amid increasing concerns by the international community.
Foreign governments and organisations fear Rohingya villages are being subject to collective punishment after an armed group on August 25 attacked police posts and an army base in Rakhine state.
The attacks - in which at least 110 were killed - were claimed by the Arakan Rohingya Salvation Army (ARSA), a group which was formed by Rohingya living in Saudi Arabia after a bout of serious communal violence in 2012, according to the International Crisis Group.
In the days following the attacks, the Myanmar army has burned down areas of Rakhine state and fired on civilians, according to rights groups and witnesses.
Scores have reportedly been killed. Al Jazeera has been unable to verify the death tolls.
While Rohingya Muslims have largely fled to Bangladesh, Rakhine Buddhists have mostly sought sanctuary in towns and monasteries to the south and east of the fighting.
"As of last night, 18,500 people have come across" from Myanmar's Rakhine state, Chris Lom, the IOM's Asia-Pacific spokesman, told the AFP news agency.
Lom said exact figures were difficult to obtain because many of those who have made it into Bangladesh might not register with local authorities.
"We also know there are people stuck at the border but we do not know how many," Lom said.
Bangladesh, which already hosts some 400,000 Rohingya who have fled Myanmar over the years, has vowed to block new arrivals and has deported some of those it has caught trying to make the crossing.
"They are in a very, very desperate condition," said Sanjukta Sahany, who runs the IOM office in the southern town of Cox's Bazar near the border.
"The biggest needs are food, health services and they need shelter. They need at least some cover, some roofs over their heads."
Sahany said many crossed "with bullet injuries and burn injuries," and that aid workers reported that some refugees "gave a blank look" when questioned.
"People are traumatised, which is quite visible."
The UN, while condemning the attacks by ARSA, has pressured Myanmar to protect civilian lives without discrimination and appealed to Bangladesh to admit those fleeing the military counteroffensive.
Northern Rakhine has been under lockdown since October last year when a previously unknown group of Rohingya fighters ambushed a series of border posts inside Myanmar.
That prompted a massive military response, leading to some 87,000 Rohingya fleeing to Bangladesh, bringing with them harrowing tales of murder, rape and burned villages.
The UN believes the Myanmar government's response to the crisis may amount to ethnic cleansing and crimes against humanity.
Satellite data recently accessed by Human Rights Watch show widespread fires burning in at least 10 areas in Rakhine.
Myanmar authorities say Rohingya "extremist terrorists" have been setting the fires during fighting with government troops, while Rohingya have blamed soldiers who have been accused of carrying out extrajudicial killings.
Source: Al Jazeera and news agencies