"Turkey, Iran, China, South Korea, Italy, and Arab states" are "far outpacing the US and UK" in winning contracts in Iraq.
French capitalists are doing well, too. "As well as investing in telecoms and building two car plants, [the French firm] Lafarge produces 60% of the cement sold in Iraq" (Financial Times, 16 December).
Shell and Exxon have won some oil contracts. But in the big Iraqi government auction in 2009: "Two of the most lucrative of the multi-billion-dollar oil contracts went to two countries which bitterly opposed the U.S. invasion — Russia and China — while even Total Oil of France, which led the charge to deny international approval for the war at the U.N. Security Council in 2003, won a bigger stake than the Americans in the most recent auction" (Time magazine, 19/12/09).
Norway's DNO International, Turkey's Genel Enerji, and the Chinese state-owned Sinopec, as well as US-based Exxon, have big oil contracts in the Kurdish autonomous region of Iraq.
Further, "the Iraqis are pursuing economic relations with countries such as Iran, which had been their enemies but are now close trading partners" (FT).
Iraqi prime minister Nouri al-Maliki publicly bemoans the lack of American companies with contracts in Iraq, but the best guess must be that Maliki (who, when in exile from Saddam, chose to live in Syria, Iran's close ally, rather than in Britain or the USA) is saying it for show.
The USA has been unable to control their outcome. The misdeeds of the US armed forces have made it harder for US companies to win contracts, and the US companies themselves more cautious about even attempting to come in.