By John Davison and Ahmed Rasheed

BEIRUT/BAGHDAD - (Reuters) - U.S.-backed militia drew within firing distance of the last road into an Islamic State stronghold in northern Syria on Thursday, part of a wave of new offensives putting unprecedented pressure on the self-declared caliphate.

The effective encirclement of Manbij by a militia called the Syria Democratic Forces is part of an assault launched last week, backed by U.S. air power and American special forces, to seal off the last stretch of Syrian-Turkish frontier.

It marks the most ambitious advance by a group allied to Washington in Syria since the United States launched its military campaign against Islamic State two years ago.

Simultaneously, Russia is backing a separate advance by forces of the government of Syrian President Bashar al-Assad against Islamic State in another part of the country.

And in Iraq, at the opposite end of Islamic State territory, the Baghdad government has sent forces to try to storm the Islamic State bastion of Falluja, an hour's drive from Baghdad.

Islamic State has also lost territory in recent weeks to Kurds in northern Iraq and anti-Assad rebels in Syria as its disparate enemies attack on a number of fronts.

But it demonstrated on Thursday it can still mount deadly attacks deep inside the territory of its foes. It claimed responsibility for two suicide bombings that killed at least 24 people in Baghdad, and was presumed to be behind a suicide bombing that killed a Western-backed rebel leader in southern Syria.

A five-year-old multi-sided civil war in Syria and a weak government in Iraq have made it impossible to wage a single coordinated campaign against the militants. But Washington and other powers hope this year will see the tide turn against Islamic State, which has ruled over millions of people in Iraq and Syria since declaring its caliphate in 2014.

SDF SEIZES ALL ROADS INTO MANBIJ

In Syria, Washington has long lacked capable proxies on the ground, but has found its first strong allies in the Syria Democratic Forces (SDF), formed last year by recruiting Arabs to join forces with a powerful Kurdish militia.

The SDF launched its new offensive last week against the city of Manbij, Islamic State's main bastion near the Syria-Turkish border west of the Euphrates River.

The overall aim is to shut the Turkish-Syrian frontier, which has served for years as Islamic State's only major route to the outside world for manpower and material, and more recently for followers returning to Europe to carry out attacks.

An SDF spokesman said on Thursday his group had reached the last road into Manbij from the west, having previously cut off supply routes from north, south and east.

"We have reached the road that links Manbij and Aleppo, from the west," Sharfan Darwish, spokesman for the Syria Democratic Forces-allied Manbij Military Council, told Reuters.

A monitoring group, the Syrian Observatory for Human Rights, confirmed that the SDF had advanced to within firing distance of the western road, positioned within a kilometer of it, and were now in effective control of all routes into the city. Civilians in the city and surrounding countryside were fleeing.

Darwish would not comment on whether the SDF was planning an assault on the city itself. He told Reuters on Wednesday forces was poised to enter, but were being cautious due to the civilian presence there.

In southern Syria, where a range of anti-Assad rebel groups include Western-backed nationalists, one of the founders of a rebel alliance called the Free Syrian Army's Southern Front was killed by a suicide bomber suspected to belong to Islamic State.

Saleem Bakour, a colonel in the Syrian army who defected to the rebels, had led rebels in battle against Islamic State fighters who pushed south after being driven out of the city of Palmyra by Russian-backed government forces in March.

"The martyr was one of the toughest leaders who fought Daesh (Islamic State). We are committed to fighting them to the end," Southern Front spokesman Issam el-Rayyes said.

BAGHDAD BOMBINGS

In Iraq, Islamic State claimed responsibility for two suicide bombings that killed at least 24 people in Baghdad on Thursday. Such bombings have become frequent again in the capital in recent weeks, after months in which security there had improved despite Islamic State's control of swathes of territory in the provinces.

The deteriorating security in the capital prompted Prime Minister Haider al-Abadi to order an assault on Falluja, Islamic State's closest bastion to the capital, two weeks ago. It began in earnest last week with troops sweeping into southern rural districts, and they entered the built-up areas of the city for the first time this week.

The Iraq assault on Falluja has the support of U.S. air power, but veers from Washington's battle plan, which called for the government to focus its forces on Mosul, Islamic State's de facto Iraqi capital further north.

Falluja, where U.S. forces fought the heaviest battles of their own 2003-2011 occupation of Iraq, has long been a stronghold of Sunni Muslim insurgents opposed to the Shi'ite-led government in Baghdad.

Washington fears a sustained campaign in Falluja could bog down the army in hostile territory and delay the recapture of Mosul.

(Additional reporting by Suleiman al-Khalidi in Amman; writing by Peter Graff; editing by Andrew Roche)