Now that U.S. combat units have disappeared from Iraqi cities, the Iraqi security forces are displaying a bit more swagger and self-confidence. The Iraqi government has become quite insistent that U.S. combat units not be seen in urban areas during daylight hours. Some American combat units sill rumble through at night, often to the consternation of Iraqi police, who thought the Americans were gone (as in, left the country.)
American and Iraqi troops never got on exactly the same page. Troops from the two countries did not have the same attitude towards professional behavior and what was effective, and what was simply adequate. U.S. troops, who had spent a lot of time training Iraqis, became resigned to accepting a lower level of performance, calling what they considered the best the Iraqis were capable of, "Iraqi good."
Some Iraqi units were capable of American levels of combat performance. It was mainly dependent on the quality of officers and NCOs. There were few of these left over from the Saddam period, because back then, only (or mostly) Sunni Arabs could fill those leadership jobs. Since then, few Sunni Arabs have been let back into those jobs. The majority Kurds and Shia Arabs just don't trust the Sunni Arabs (currently 15 percent of the population, although they were 20 percent in 2003, and many Sunni Arabs still believe, against all reason, that they are the majority in Iraq.)
The Iraqi security forces have learned, via hard won experience, how to do what has to be done. They can man checkpoints, search for bombs and carry out raids and search operations. The Iraqi soldiers and police know that all this can be risky duty. But unlike Americans, the Iraqis tend to be more fatalistic about the consequences of being sloppy. This disturbed their American (and other Western) trainers, but old hands learned to accept it. That's how things are in this part of the world, and why the colonial era British observed, over a century ago, "you can't hustle the east."
Of course, the many Iraqis who immigrated to the United States, and joined the armed forces, learned their lessons as well as anyone else, and are indistinguishable from other American troops. Until they turn around while on patrol, and begin speaking in Iraqi accented Arabic. This catches Iraqis unawares, seeing an American soldier, who is acting like an American soldier, suddenly speak like an Iraqi. How are such things possible? That's what American military and police trainers would like to know. But many have come to accept that it's the culture.
Some Iraqi military and police commanders agree, with their American counterparts, that there are some serious problems with the capabilities of many Iraqi units. The Americans are much more flexible and precise. U.S. troops can deal with a nasty terrorist situation better than Iraqis (who tend to just shoot everyone in sight, and not show much discipline at all.) If pushed too hard by terrorists or militias, those same Iraqi commanders fear that the troops and police will simply flee. That has happened many times since 2003. And before that, Saddam had to take severe measures (summary execution was a favorite) to maintain the discipline and steadfastness of some military units during the 1980s war with Iran.
The Iraqi military has long been considered the most inept and ineffective in the Middle East. That's pretty bad, because the region is notable for the shabby track record of its military units. But many Iraqis, and some Americans, believe that six years of training by Westerners, and working closely with American troops, has changed some of the basic attitudes. There's some evidence of that, but it will take a year or more of operating on their own, before the Iraqi leadership, and the troops, are sure.