What is the first habitat that you think of when you hear the word turtle? Yes, the water. Turtles are aquatic reptiles that need to have a water environment to live in. However, no, turtles cannot breathe underwater. You will see them creating bubbles underwater as they breathe out, and that can lead people to think that they breathe underwater. That is not true. They breathe OUT while underwater, they still need to get to the surface of the water to breathe in.
There are some turtlespecies that can absorb oxygen directly from the water, but they still need air. That is why, unlike fish, it is imperative to give them a base of land in their aquarium for them to climb onto. Swimming non-stop will cause them to be fatigued, and eventually die. Furthermore, if they can not get to the surface for any reason, they can also drown.