A few years ago I repeatedly experienced the same symptoms as you while while towing a small car with my class A gas rig. I thought my front wheels were about to fall off! I had the front end of my class A MH checked and found some worn components that were replaced. This fixed the problem for a while.
I traded the MH for a new DP, then about 6 months later, I began having the same problem. Fortunately the new MH had a rear-view camera and I could see the shaking was coming from the towed car. It turned out that the front struts on the car were worn out and were unable to quickly stop the roll oscillation that was created when finishing a turn. It was caused by some interaction between the car body rolling in the turn, small normal steering changes that happen when all cars experience body roll, and the normal lateral play in the hitch and tow bar.
After having the car's front struts replaced there has been no repeat of the problem for about 2 years. Before I had the car's front struts replaced, I was able to prevent the shaking by allowing the turn to complete and the car to straighten out before accelerating, but like your situation once the shaking started I had to come to nearly a complete stop to make the shaking abate.
While troubleshooting I would not overlook defects in the car, the attachment and integrity of your adapter plate, tow bar, and the hitch on your MH.
If you find no problem with the above items and you can verify using your rear-view camera that the problem is with the car, You might check the angle of the tow bar to the ground when towing and on level ground. Depending on the model of Blu Ox tow bar you have, I believe they recommend that the bar sits level or slightly raised in the front.(at the MH receiver) check your tow bar manual to be sure. If the angle is off quite a bit,it seams to me that body roll could be induced in addition to the normal roll caused by centrifugal force created in the turn, This additional body roll might be enough to start the chain of events that cause the shaking.
Dave