Is it normal to feel like I am breathing fast in the 3rd trimester?

Depends. As the baby grows, it presses up on the lungs and certain muscles in the ribs relax and you may feel like you are taking shallower breaths more often. But if this happens suddenly or you feel like you are really struggling, you need to see someone right away. Blood clots in the lungs are also more common in pregnancy and can be very serious.
Yes. The weight of the unborn fetus and hormonal changes associated with pregnancy can increase your breathing rate/respiratory rate in the last trimester.