Yes, and it could be worse because private jets are usually smaller than commercial airplanes. Turbulence has to do with the air, not the plane, and small aircraft are affected a lot more by lesser turbulence. I can positively state that small aircraft are affected a lot more by lesser turbulence, and every tiny lake and water hole had us climbing up and down by 50 feet.
During my long cross‑country requirement, flying a Cessna 152, I flew between 2 peaks and hit a downdraft that I couldn’t climb out of, and any commercial plane that did that would have screaming passengers. Private jets typically have about half the wing loading of the big commercial aircraft, which makes them more responsive to turbulence and a bumpier ride when they’re going through it, but their higher cruise altitudes keeps them out of it more.
Some private jets do fly at higher altitudes, sometimes up to 50,000 feet as opposed to 30–40,000 feet for a commercial jet, so you might experience less turbulence, but there will still be some. This happened when the business jet inadvertently flew underneath an A380 and entered its wake, and it was not a fun experience for everyone onboard.
Simply put, the bigger/heavier the airplane, the stronger the vortex, and airplanes can easily tumble out of control once they enter the vortex of another airplane. These vortices begin from the moment the aircraft takes off and go on all the way until it comes to a complete stop after landing. This is also why airplanes generally try to steer clear of each other.
Most airplanes aren’t designed to do the kind of acrobatics that the Challenger was subjected to, and basically anything that is not built like a fighter jet would get thoroughly wrecked if it is forced to perform such violent maneuvers. The Challenger also got lucky it happened way above at 34,000 feet. Had it occur close to the ground as it was going to land or just after takeoff, it’s very likely the business jet would have crashed. The idea is that the airplane trailing behind or crossing behind the previous airplane should wait until the vortex has dissipated enough so that they wouldn’t get thrown around like this Challenger jet.