The best information becomes easier with pictures.
Here we see the start of a turn. The rider is reasonably straight but the inside shoulder is dropped and the head turned. This causes the whole inside of the rider to fall inward and notice the hand has done that also. This now caused the rider to pull the head into the circle and the horse as a result has fallen on the inside shoulder.
Compare the above picture to this one following where the rider is equally placed in the saddle, the head is not turned and the riders hands are where they should be. The horse is properly flexed (notice there is very little bend which is correct).
On a 20 meter circle on a well trained horse, no, you don't need to turn your shoulders.
On a green horse who's having troubles, or on a sharper turn, yes, you should turn your shoulders a tad. You should NOT drop your inside shoulder when turning, that is incorrect and will result in a rushed turn/circle or bulged outside shoulder on the horse. Your shoulders need to stay up and back, over your center of gravity.