Welcome! KJ/KK is the body style. JK is the Wrangler, WK is the Grand Cherokee, etc. KJs were built from 2002-2007 and have the rounded headlights. KKs are from 2008-2012, which is what you have.
That hopping behavior is normal for a 4x4 vehicle when in 4wd and is called crow hop. If you do not have a full-time transfer case (your 4x4 shifter will say 4lock instead of 4auto if you don') the transfer case locks the front and rear driveshafts together so they spin at the same speed. When you turn, the front wheels turn at a different speed than the rear, so the driveline starts to bind until the tires break traction which is the bump you feel. Here is Jeep's FAQ site with some more helpful info:
http://www.jeep.com/en/4x4/faqs/This shouldn't happen in 2wd with stock tires/suspension. You may have a loose tie rod or ball joint thats on its way out.
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