Explosions throw up clouds of smoke near Iraqi soldiers with armoured vehicles, who check for casualties, spot enemy forces in the desert ahead of them and then open fire.

But the blasts are only simulated artillery fire and the "enemy forces" are pop-up silhouette targets, not gunmen and vehicles.

That is just as well for these soldiers on a training exercise, as the puffs of dust kicked up by bullets downrange indicate that their aim is often off the mark.

An Iraqi company commander led his soldiers in the exercise at the massive Besmaya military base southeast of Baghdad, but it was a foreign contractor who controlled the scenario.

It was the contractor who ordered the targets raised and lowered amid the sounds made by .50 calibre machine guns on the armoured vehicles and the chatter of M-16 rifles.

Contractors, who also assist soldiers in preparing for drills and with after-action reviews, are at the forefront of US efforts to train Iraqi forces.

Negotiations on a post-2011 US military training mission broke down last year over Iraqi reluctance to offer the trainers immunity from prosecution, and almost all American soldiers left the country last December.

But they left behind the Office of Security Cooperation-Iraq (OSC-I), a group of about 157 military personnel under US embassy authority, and some 600 civilian contractors, mostly retired soldiers.

They are working with the Iraqis on everything from training on new equipment, such as US M113 armoured personnel carriers and M1 Abrams tanks, to military education.

"This is the model that Iraq is very comfortable with -- low US presence in uniform, a lotta contractors, and they're getting the quality instruction, the quality training that they really need," Lieutenant General Robert Caslen, the chief of OSC-I, told AFP at Besmaya.

OSC-I is an organisation that was going to exist anyway, and similar offices exist in other countries with which Washington has deep military ties, but it took on increased importance as a result of the pullout.

The trainers have been consistently reluctant to provide estimates of the troops trained or target figures.

"Since you're the only military force in town and you have the ability to provide the ... security training and the security education that Iraq is looking for, you tend to be in demand," said Caslen.

The US military personnel in OSC-I are mainly involved in "bringing the equipment in, working all the financing, working all the regulations, the policies and things like that," Caslen said, though some also make sure security measures are in place and oversee training.

Iraq has placed orders for more than $10 billion (7.6 billion euros) in military equipment and assistance, the Special Inspector General for Iraq Reconstruction, a US watchdog, said in a quarterly report on April 30.

"Overall, we want to have Iraq ... provide the security necessary to defend against external threats and defend against internal threats within the country, to provide security for the Iraqi people," and "for Iraq to be a responsible partner within the region," Caslen said.

"Our main objective is to provide the fielding (of equipment), to provide the training ... on the equipment as they receive it, to train with the equipment, then to professionalise Iraqi security forces," he said.

Providing large-scale training to Iraqi forces in the absence of a larger US military force in Iraq presents a challenge, but OSC-I has come up with various ways to try to do it.

Caslen outlined a battalion training programme under which "the staff comes in for a week ahead of time before they actually execute this 60-day training plan, so we train the staff, the battalion commander and the staff, on military decision-making and things like that," he said.

"And then we're working with CENTCOM (US Central Command) on joint and combined training exercises in the region," he said, noting that there will be an exercise in Jordan involving the Iraqi air force and Iraqi counter-terrorism forces in May.

"And finally, you see education taking place not only in Iraq, but you also see military education for all echelons taking place even in the United States and some other nations as well," he said.

Asked about the main things Iraqi forces need to work on, Caslen said: "They're on the path right now to develop the capabilities to deal with external type of threats wherever they may come from."

Both Iraqi and US personnel, including Caslen, have said external defence is a weak point for the Iraqi military, which has been mainly focused on quelling internal violence.

Caslen also noted that "Iraq is very concerned about defending their airspace," but that the F-16s it has ordered from the United States will not be up and running for several years, leaving a "gap in the middle."

And while he said that "they're making tremendous strides in the counter-terrorism capabilities," he noted that some problems with intelligence capabilities remain.

"They can kick a door in as good as anybody," Caslen said, "but they have some intelligence capability gaps, so the issue is, do they know exactly what door they need to kick in."