More than 55% of Iraqis voted in parliamentary election despite attempts by Sunni Islamist insurgents to disrupt the landmark vote with attacks that killed 38 people.
Preliminary results were not expected for another day or two.
'It (turnout) is between 55 and 60%,' said Hamdiya al-Husseini, a commissioner of Iraq's independent electoral commission, or IHEC.
Prime Minister Nuri al-Maliki's State of Law list claimed it was on course for victory in Baghdad and Iraq's Shia south, a claim that could not be verified but which, at least in the south, appeared to be backed by informal, early vote tallies.
'The State of Law Coalition list is leading among other lists in Baghdad and other southern provinces,' said Ali al-Dabbagh, government spokesman and State of Law candidate.
Lawmaker Haider al-Ebadi, a State of Law candidate and member of Mr Maliki's Dawa party, said initial results suggested the coalition was ahead in ten provinces.
'In Baghdad and south of Baghdad, the State of Law was number one. But the special voting and voters abroad, this has not been concluded yet and could alter the outcome,' he said.
There were 250,000 voters abroad, he said, compared to expectations that more than 1m Iraqis might vote overseas.
Most Iraqis abroad are believed to be minority Sunnis and their votes could be crucial for the chances of a secular, Shia-Sunni alliance headed by former premier Iyad Allawi.
Mr Maliki faces a stiff challenge from his former Shia Islamist allies grouped in the Iraqi National Alliance.
The powerful Supreme Islamic Iraqi Council, which is part of that bloc, said the vote appeared evenly split between Mr Maliki and INA in early counting.
Mr Allawi's bloc, Iraqiya, was running third, ISCI said.
Thaer al-Naqeeb, an Iraqiya candidate and close aide to Allawi, said results were not clear so far but initial figures put Iraqiya ahead in the northern and western provinces.
Iraqiya got between 70-90% of votes in those provinces, he said.
In Iraqi Kurdistan, a new party was challenging President Jalal Talabani's Patriotic Union of Kurdistan, one of two groups that have dominated Kurdish politics for decades.
