The rigid shell enclosing the body presents a special problem for most turtles. ( some turtle species are equipped with a muscle called Muscularis striatum pulmonare which partially covers the lungs and presumably assists in respiration )
Lungs are the only compressible or expansible organs within the shell, and can be filled or emptied from air forcibly by withdrawing the head or legs.
The only way of active breathing is by flexing the membrane situated across the rear part of the shall behind the lungs. Using abdominal muscles, a turtle inhales by pulling the membrane back and axhales by pulling it forward. This action compresses or expands the lungs, and it can be seen by observing the area of soft skin around the base of the hind legs being pumped in and out during breathing.
A turtle in water with only its nostrils out, exhales passively using hydrostatic (water) pressure, and only has to inhale actively.

As one of the means of defense turtles stay hidden in their shell. Unable to breath the regular way, they are exposed to high CO₂ ( carbon dioxide ) levels in their lungs, sometimes for long periods of time. However, unlike most other vertebra, turtles have adopted to this inconvenience by being able to increase or decrease the pH level ( acidity ) of the blood ( blood can contain more CO₂ per volume at some pH levels ).
Another factor helping turtles is the ability to empty their lungs more completely than most other vertabrates. They also breath more rapidly and deeply than usually required - a process called hyperventilation.
Inhaled oxygen is normally stored in blood haemoglobin, but in turtles a significant amount is stored myoglobin in the muscles.
When turtles are active glycogen in the muscles is converted to lactic acid which is toxic, and has to be burned off with oxygen. In turtle this is done in a process called anaerobic glycolysis, a process that doesn't require free oxygen. ( possibly due to using the oxygen stored in myoglobin ).
Therfore, turtles are able to exist for prolonged periods of time without respiration, in environments with little or no oxygen.
This lets the terrestrial turtles to wait until the danger is gone with their head and limbs retracted, and allows an aquatic turtle to dive and remain submerged for long periods of time.
This ability, combined with the phenomenon of slowing the heartbeat rate during a dive or in lower temperatures, enables turtles to hibernate during the cold seasons.
According to Dr. Pritchard's book, the observed pulsating of the throat in turtles has nothing to do with the breathing process, but is simply " sampling " of the environment by pumping air over the olfactory surfaces.