Here is my favorite youtube video.
The pivot is forward motion. You point your feet in the direction you want to go and "move your horse's eye". You don't want to be at your horse's shoulder, but rather even with his eye and give him room to turn. Biggest thing is to watch at the bottom of your peripheral vision that the front leg is crossing over during the turn. The back end should follow.
To square-up get your rear feet in position and then work to even up the front. Works best to walk in hand, halt and then position your horse. Push up and back (hard to explain so bear with me) on the lead when positioning the back feet and pull down and back for front feet.
I would suggest that you explore the youtube videos and watch the body positions and mannerisms of the handlers.
Hope this helps!