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A pet snapping turtle usually will not bite unless it is scared by rough handling or injured in some way. Careful handling is the key to preventing bites. If you are bitten, a turtle can be removed by being calm and placing the turtle in water until he lets go.
Removing the Turtle
If a turtle has bitten you, do not attempt to pull or pry the turtle's mouth off of you. This will worsen the damage to your skin and will frighten the turtle or cause him pain. Submerge the turtle into water. Once submerged, hold still and he will let go.
Once he has released his grip, move away from him quickly and give him time to calm down if possible. If you need to remove him from the water, be sure to grasp the back of his shell and hold him away from your body. Be careful not to drop him and cause injury.
Under no circumstances should a snapping turtle that has bitten you be killed. Upon death, a turtle's jaw clamps down with even more pressure, which will cause you more injury. And a dead turtle cannot release his grip.
Treating the Wound
Since turtles are known to carry salmonella, any bite that breaks the skin needs to be treated by a medical professional. Antibiotics may be needed, as well as thorough cleaning of the wound. If the skin is not broken, wash the bitten area with warm water and plenty of soap.