Not all animals are restricted to just one single breathing vessel like we humans and most other animals do. It is true that turtles breathe through their back orifice. However, not all turtles can do this trick. A few species such as the Fitzroy River turtle can pull this one. By ‘breathe’ it means actual breathing, taking in air into the body, as opposed to taking it out (farting), which perhaps also features since breathing has to be two-way. This area is still yet to be well researched on.
This is not the primary mode of going about it but it sure comes in handy for the turtle species that do it this way. They breathe using their lungs, just as most other animals but they have also been endowed with a pair of sacs that are located within their ass which can take in water and sieve the oxygen from it. This happens when they are submerged in water.
The art of breathing through the anus is called cloaca respiration and is best established in pleurodires.