While diving at Rich's canyon, a wall dive, we encountered a Hawksbill turtle and had the opportunity to swim with the turtle for quite some time. As we tell divers, when encountering a turtle, we let the ladies swim first as the turtle can tell the difference from a female and male diver and are less intimidated by the female divers.
We recommend not touching the turtle or other large marine creatures as you may scare or provoke them. When within about 10 feet of the animal, it is best to swim parallel to the animal otherwise it is likely concerned it is about to be attacked. Swimming parallel raises the animal's curiosity and it will likely stay around to investigate the divers.