At least 27 people, including children, were killed in the attacks.
They are the latest in a relentless wave of killing that has made for Iraq's deadliest outburst of violence since 2008, with 980 people killed in the last month.
The mounting death tolls are raising fears that the country is falling back into the spiral of violence that brought it to the edge of civil war in the years after the 2003 US-led invasion.
Sunday's blasts began about 9.30am in the Shiite Turkomen village of Qabak, just outside the town of Tal Afar. The area around the stricken village has long been a hotbed for hard-to-rout Sunni insurgents and a corridor for extremist fighters arriving from nearby Syria.
One car bomb in the tiny village targeted an elementary school while children aged six to 12 were in class as another struck a nearby police station, Tal Afar mayor Abdul Aal al-Obeidi said.
The dead included 12 children, the school principal and two policemen. Another 90 people were wounded, he said. The village is home to only about 200 residents and part of the single-storey school collapsed as a result of the blast.
"We and Iraq are plagued by al-Qaeda," Mr al-Obeidi said. "It's a tragedy. These innocent children were here to study. What sins did these children commit?"