A car bomb exploded outside St George’s Catholic church in southern Baghdad just before 6.30pm (1530 GMT), followed minutes later by a second outside St Matthew’s church.
Victims from both blasts, some carried by injured friends or relatives in torn and bloodstained clothes, were rushed to Yarmuk hospital.
A doctor said at least three people were killed and 40 injured.
A few hours later, a car bomber ploughed into four police cars parked outside the hospital entrance, killing at least five policemen, police said.
Several more explosions echoed across the city later in the night, but there was no immediate word on casualties.
The wave of bombings swept Baghdad as US marines began their full-scale offensive to capture the besieged city of Falluja, 50km west of the capital.
In the latest of almost daily bombings on the main road to Baghdad airport, at least three people were killed in a car bomb attack on a checkpoint earlier in the day.
Three people were killed in an
Fighters battling Iraq’s interim government and its US backers have stepped up attacks around the country since US forces began building up for their assault on Falluja, seen as the centre of anti-US fighters.
Iraq’s Christian minority has also been hit. Five churches were struck in a string of bombings in October. Eleven people were killed in similar attacks in August.
Iraq’s 650,000 Christians, mostly Chaldeans, Assyrians and Catholics, comprise about 3% of the population.