* Rate up from 2,000 to 3,000 a day late last year
* UNHCR says could surpass 1 million well before June
* Water shortages worsening in Syria, UNICEF says (Adds details, quotes)
By Stephanie Nebehay
GENEVA, Feb 8 (Reuters) - About 5,000 refugees are fleeing Syria each day, seeking safe haven from war and its devastating impact on basic living conditions, the United Nations said on Friday.
An acceleration in the exodus - up from 2,000 to 3,000 late last year - means the total could exceed one million refugees well before the end of June, its previous forecast, the U.N. refugee agency said.
"This is a full-on crisis," Adrian Edwards, spokesman of the U.N. High Commissioner for Refugees (UNHCR) told a news briefing in Geneva. "There was a huge increase in January alone, we're talking about a 25 percent increase in registered refugee numbers over a single month."
About 150,000 Syrians poured across Syria's borders in January, most into Jordan and Lebanon which have seen a massive increase in the inflow.
Since the conflict began two years ago, more than 787,000 Syrians have registered as refugees or are awaiting processing in the region, mainly in Lebanon, Iraq, Jordan and Turkey. The total includes 15,000 in Egypt, where the U.N.'s World Food Programme began this week to distribute food vouchers.
"We are working around the clock to keep up with the needs and demands of the refugees," UNHCR spokeswoman Sybella Wilkes said.
The UNHCR, in a nearly $1 billion funding appeal in mid-December, forecast that there could be up to 1 million Syrian refugees by the end of June.
"At the rate that refugees are arriving, we can expect to surpass the one million mark months before," Wilkes said.
Turkey has spent more than $600 million sheltering refugees from the conflict, Finance Minister Mehmet Simsek said. More than 177,000 Syrian refugees are in 16 camps with thousands more staying with relatives or in private accommodation.
President Bashar al-Assad's forces fought back on Friday in an effort to retake sections of the Damascus ring road from rebels trying to tighten their noose around the capital, opposition activists said.
Around Syria, water shortages are worsening and supplies are sometimes contaminated, putting children at increased risk of diseases, the U.N. Children's Fund (UNICEF) warned on Friday.
The agency's first nationwide assessment revealed that water supplies in areas affected by the conflict are one-third of pre-crisis levels, UNICEF said.
"It points to a severe disruption of services, damage done to water and sanitation systems, and limited access to basic hygiene, all of which puts children at a much increased risk of disease," UNICEF spokeswoman Marixie Mercado told the briefing.
Six places were identified as most at risk - rural Damascus, Idlib, Deir al-Zor, Homs, Aleppo and al-Raqqah.
Dirty water diseases including Hepatitis A are spreading in Syria, compounding the problems of hospitals that are perilously short of medicine and doctors, the World Health Organization said on Tuesday.
"Interviews conducted with recently arrived refugees in Zaatari camp in Jordan said that a large majority of households were reporting cases of diarrhoea among children and adults, as well as scabies and head lice," Mercado said. (Reporting by Stephanie Nebehay; Editing by Oliver Holmes and Robin Pomeroy)