More than two million Syrians are now registered as refugees, after the total went up by a million in the last six months, the UN's refugee agency says.
More Syrians are now displaced than any other nationality, says the UNHCR.
France and the US are continuing to push for military action over alleged chemical weapons use by Syrian forces.
There are suggestions that President Barack Obama may be planning much wider action than the limited strikes that have been publicly proposed.
The reports emerged as senior US politicians were set to speak before a congressional committee, to rally support before a vote expected next week on whether the US should launch military action.
Tensions remain high in Syria and the surrounding region.
Russia said on Tuesday that it had detected two ballistic missiles being launched towards the eastern Mediterranean coastline, sparking speculation of a connection to the Syria crisis.
But Israel later confirmed that it was a joint US-Israel missile test.
The BBC's Richard Galpin in Jerusalem says tests like this are usually planned long in advance, but it is still a sign that the Israeli military is taking very seriously the possibility that US air strikes on Syria, if they do happen, could lead to retaliatory attacks on Israel - either by Syria itself or by its ally, the Shia militia Hezbollah in Lebanon.
The UNHCR said in a statement on Tuesday: "Syria is haemorrhaging women, children and men who cross borders often with little more than the clothes on their backs."
Around half of those forced to leave are children, UN agencies estimate, with about three-quarters of them under 11.
Just 118,000 refugee children have been able to continue in some sort of education, and only one-fifth have received some sort of counselling, with agencies warning of a "lost generation" of child refugees ill-equipped to help rebuild Syria in the future.
Lebanon has received the highest number of refugees, at 700,000, even though it is the smallest of Syria's neighbours and one of the least able to cope.
There is now thought to be one Syrian refugee in Lebanon to roughly every six Lebanese. Jordan and Turkey have taken in the second and third highest numbers respectively.
As well as those who have left the country, a further 4.25 million have been displaced within Syria, the UNHCR says, meaning that more people from Syria are now forcibly displaced than from other country.
Pointing out that more than 97% of Syria's refugees are being hosted by countries in the surrounding region, the UNHCR said the influx was "placing an overwhelming burden on their infrastructures, economies and societies".
It appealed again for "massive international support" with the crisis.
International aid agencies are struggling to cope, having only 47% of funds required to meet "basic refugee needs", the UNHCR says.
"It took two years to reach the first million refugees. It took six months for the second million to be reached," the UN High Commissioner for Refugees, Antonio Guterres, told the BBC.
He said officials could envisage three million refugees by the end of 2013.
One of the biggest single waves of refugees occurred in mid-August, when thousands of people from north-eastern Syria poured over the border into Iraqi Kurdistan.
Iraq has the fourth largest population of Syrian refugees, with over 170,000.
The UN says it is the worst refugee crisis for 20 years, with numbers not seen since the 1994 genocide in Rwanda.
More than 100,000 people are thought to have died since the uprising against Syrian President Bashar al-Assad began in March 2011.
'Deter and degrade'
The US and French governments are pushing for a military strike in reaction to what they say was a chemical weapons attack carried out by Syrian government forces on the outskirts of Damascus on 21 August.
The US has put the death toll at 1,429, including 426 children, though other countries and organisations have given lower figures.
US Defence Secretary Chuck Hagel, Secretary of State John Kerry and Joint Chiefs of Staff Chairman Gen Martin Dempsey are due to appear before the Senate Foreign Relations Committee later on Tuesday.
They are trying to muster support for military intervention in Syria in the run-up to a vote in Congress expected next week.
The document seeking congressional backing speaks of sending "a clear signal of American resolve", aiming "to deter, disrupt, prevent and degrade the potential for future uses of chemical weapons or other weapons of mass destruction".
Many analysts had assumed the strikes would be fairly limited in scope.
But Gen Jack Keane - a retired US Army vice-chief of staff who spoke to key Republican senators after a White House briefing on Monday - told the BBC they had been given the indication that strikes would go far beyond targeting Syria's alleged chemical weapons.
"He's going to deter and degrade - and the important word is degrade - significant military capability of Assad's regime," said Gen Keane.
Senators had been encouraged by "the scale and robustness" in President Obama's plans, he said.
Mr Obama already appears to have won the support of two of his fiercest foreign policy critics, Republican Senators John McCain and Lindsey Graham.
Mr McCain told reporters on Monday that a vote against strike action "would be catastrophic in its consequences" for US credibility abroad.
Qatar, thought to be the main financial backer of the Syrian opposition, has also expressed support for the proposed strike.
Qatari Foreign Minister Khalid Bin Mohamed Al Attiyah told the BBC that President Assad would kill every Syrian if he remained in power.
"It is not about regime change. It's about the people who've been killed every day and we would like to see the people protected."
He said he would ideally have preferred to see an Arab intervention, but the Arab League had failed to act.
In France, a report presented to parliament by Prime Minister Jean-Marc Ayrault on Monday said the 21 August attack involved the "massive use of chemical agents" and "could not have been ordered and carried out by anyone but the Syrian government".
But in an interview with French newspaper Le Figaro, Syrian President Bashar al-Assad denied being behind the attack, saying it would have been "illogical".
He warned that foreign military action could ignite a wider regional conflict.
Nato Secretary General Anders Fogh Rasmussen also said on Monday that be believed the Assad government was behind the attack.
He did not expect Nato to be involved in any action, but said there must be "a firm international response" to deter any future use of such weapons or else it would send a "dangerous signal to dictators all over the world".
The head of Germany's security service has told MPs there that it, too, believed the Assad regime carried out a chemical attack, and that Sarin was the gas used.
Syria is known to have extensive supplies of chemical weapons.