The shootings and bombings were the latest in months of unrelenting bloodshed, Iraq's worst since 2008, and come despite wide-ranging operations targeting militants and tightened security.
In northern Nineveh province, a suicide bomber drove through a police checkpoint and detonated a car rigged with explosives, killing two policemen and wounding three others and a civilian who was nearby, security and medical officials said.
The bombing struck the Shiite Turkmen city of Tal Afar.
In south Baghdad, a roadside bomb targeting a police patrol killed a police major and wounded three policemen, officials said.
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Violence so far this year has left more than 5,450 dead, the country's worst violence since 2008, and Prime Minister Nuri al-Maliki has appealed to the US for greater cooperation in the fight against militancy.
In addition to major security problems, the Iraqi government has failed to provide adequate basic services such as electricity and clean water, and corruption is widespread.