Is it common for my heart to race in pregnancy?

Yes: There are several reasons to explain this common occurrence. Anemia is common and it means you have less oxygen in your blood. Because your oxygen needs have stayed the same or possibly increased, you body is going to ask your heart to work harder to make up for the lower oxygen (really hemoglobin). Another reason I have found is low magnesium. Consider 200-400mg of magnesium glycinate.

Pregnancy heart: The heart rate is normally faster during pregnancy (from 4th month onwards) than in the non-pregnant state. You should feel your pulse or heart beat and count how many beats you have in one minute during those times you have the sensation of racing heart beats. Discuss your feelings and findings next time you see your doctor.