Farting is normal. Carrots are bad and potentially fatal, if your bunny eats too many of them. The photo of your bunny shows the bottom of her back feet, and they are clean. If her butt is also clean and if you have not seen any cecotropes then she is probably going to survive this episode. If you have been seeing uneaten cecotropes or if her but is messy, you should get to a good rabbit vet as soon as possible.
Farting is a normal part of being a bunny. See related question Can rabbits fart?
Carrots are not healthy for rabbits see related Are carrots a healthy part of a rabbit diet?
Also related Do bunnies eat their droppings? & Bunny suddenly kicking litter and stinking