No! It definitely happens and is considered a normal part of the process.
There are lots of things that you might worry about when having your baby: how will I know I am in labor? Will it hurt? How long will it take? Having a bowel movement while pushing out your baby should not be one of them. This is a very common thing that obstetricians, midwives, and nurses are used to and see every day. In fact, it can be reassuring to see because it means that you are pushing the “right” way; that is, you are directing your forces effectively. This makes sense since the pelvic structures all sit right next to each other and share many nerves.
If you think you can avoid this by using an enema or taking castor oil (or something related) on the way to the hospital, you may want to consider skipping those steps. Those medications may just leave you feeling nauseated and give you diarrhea, which is exactly what you were trying to avoid.