Baghdad (AFP) - The US posted tens of thousands of troops in Iraq, huddled with its leaders and helped craft its laws -- but with the country swamped by deadly protests, Washington is staying out of the fray.

Its apparent absence during a key turning point in Iraq lays bare how much its interests and influence have waned since the 2003 US-led invasion that opened the door to fellow Shiite-majority neighbour Iran.

"The (US-Iraq) gulf has never been so big, and keeps getting bigger," a senior Iraqi official told AFP on condition of anonymity.

After the invasion, the US effectively dismantled and rebuilt the Iraqi state, ushering in a new class of political elites with whom it crafted close personal links.

It trained a new military, deploying more than 170,000 troops to Iraq at the peak before withdrawing in 2011.

Since then, American soldiers have helped Iraq defeat jihadists, and US officials conferred closely with their counterparts on the 2017 Kurdish independence referendum, the 2018 parliamentary vote and the ensuing cabinet formation.

Now, protesters across Baghdad and the Shiite-majority south are demanding an overhaul of the US-crafted system, but the US has remained comparatively restrained.

It has issued a half-dozen statements condemning violence but stopped short of using its diplomatic muscle to resolve the crisis.

In the past, Washington would have been "much more overt," the top Iraqi official told AFP.

"The US back in 2003 shaped this current Iraqi government structure, which delivered this political class," he said.

"Do they want to be engaged in rectifying it? I think the jury is still out."

Secretary of State Mike Pompeo, addressing reporters in Washington on Monday, insisted that the United States was "a force for good" in Iraq and would use sanctions against anyone found stealing wealth or killing or wounding peaceful protesters.

"We support the Iraqi people as they strive for a prosperous Iraq that is free of corruption and Iranian malign influence," he said.

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- Fraying friendships -

"The bottom line is that the US state-building project in Iraq has failed," said Kirk Sowell, an analyst who writes the Inside Iraqi Politics newsletter.

Since protests erupted on October 1, more than 330 people have died, authorities have imposed an internet blackout, and activists have been threatened and kidnapped.

Last week, Pompeo telephoned Prime Minister Adel Abdel Mahdi and "deplored the death toll," but four protesters were killed the next day.

Perhaps most worrying for the US is the role of Major General Qasem Soleimani, Iran's pointman for Iraq, in brokering deals among political forces in Baghdad.

"US influence is not really zero, but it is negligible during the current crisis," said Sowell.

That is partly because Iraq has filled out its own institutions, and US troop numbers have drastically dropped, said Robert Ford of the Middle East Institute.

Ford was a diplomat at the sprawling US embassy in Iraq between 2004-2006 and 2008-2010.

But the mission now sits mostly empty after an ordered US withdrawal in May, as tensions rose between Tehran and Washington over the former's nuclear ambitions.

"That in and of itself shows US interests are reduced," he told AFP.

There is also little shared history between current Iraqi officials and the administration of President Donald Trump, unlike with previous US governments.

"I don't think President Trump could pick up the phone, talk to Abdel Mahdi and rely on old times together or face-to-face meetings," said Ford.

- 'Kiss of death' -

In fact, Iraqi and US officials say ties between the White House and the premier's office are at their "coldest" since 2003.

They say the White House has postponed a bilateral meeting at least three times because it was "angry" Abdel Mahdi was not distancing himself more from Iran.

But among a political class with deep, decades-old ties to Iran, Abdel Mahdi is "probably the best we could hope for," a senior State Department official said.

Tehran and its Iraqi allies, including armed groups, depict any party seen as close to the US as a "conspirator" seeking instability, making it politically costly to cosy up to Washington.

"Iraqi actors used to want others to know they had access to the US. Now, it's the kiss of death," said Ramzy Mardini of the United States Institute of Peace.

That logic also applies to the current anti-government demonstrators, which Iran-backed parties have sought to paint as US-backed "agents."

Western officials in Baghdad told AFP they were wary of signaling open support for the protesters because of such claims.

Demonstrators have directed their ire at the governing political class but also on perceived Iranian overreach, a dimension Washington has welcomed without explicitly backing the rallies.

Direct criticism of the US, surprisingly, has been rare -- even though it was the main architect of the system.

That could change if rallies continue to be met with violence.

"The legacy for the younger generation is that it will see the US put out talking points, but not take action," said Mardini.

"It makes it harder for US policymakers to regain the trust of the future political class."