Does Bella Cullen turn into a vampire after Edward bites her during Renesmee's birth in Breaking Dawn by Stephanie Meyer?

Yes, Bella does turn into a vampire following Renesmee's birth. Throughout the novel, we see how the growing baby affects Bella on the outside. Her health rapidly deteriorates because she is a human carrying a baby we can assume is at least part vampire. The baby's supernatural attributes are too much for a human like Bella to withstand. She loses weight, needs to drink blood, and shows signs of physical exhaustion.
During Renesmee's birth, complications arise. Bella is dying, and Edward decides to use his bite or saliva to mend Bella's wounds. Following the birth, Edward's bite is able to heal Bella and turn her into a vampire. Faced with the possibility of losing Bella, Edward makes a choice that allows Bella to stay in his life. Despite his original reluctance to "turn" Bella, he changes her into a vampire to prevent her human death.