Yes, sea turtles can drown as they have lungs just like other reptiles and similar to our own lungs. Sea turtles cannot breathe underwater, however they can hold their breath for long periods of time. However, a stressed sea turtle, such as one entangled in a ghost net for example, will deplete oxygen stores rapidly and may drown within minutes if unable to reach the surface.
Sea turtle drownings have been documented when turtles became caught in active fishing nets or ghost gear. Typical signs of drowning include a comatose state, lack of reflexes, water in the lungs and specific tissue alterations in the lung, which are visible on radiographs.
Not all turtles die immediately and while still in the water. Once turtles are comatose, they have about a 50 % chance of recovery.
References:
Norton TM 2005. Chelonian emergency and critical care. Seminars in Avian and Exotic Pet Medicine 14 (2): 106-130. Pointer IR and Harris ANM 1996. Incidental capture, direct mortality and delayed mortality of sea turtles in Australia’s Northern Prawn Fishery. Marine Biology 125: 813-825.