Remember though that Anakin was first assigned Ahsoka in the first place so he could learn how to deal with his problems with emotional attachment. A Jedi Knight naturally sees his or her apprentice move on when they are promoted to knighthood. This situation is very different indeed. Ahsoka was kicked out of the Jedi...and wrongly so...plus Ahoska revealed to Anakin that she knows about him and Padme. This is a bond that Anakin won't fast forget. They've a bond that is just as strong as his with Obi-Wan's or his and Padme's.
Palpatine is slowly isolating those closest to Anakin. Separating him from his Padawan Learner would be an absolute priority. Besides which he never "turned" on Ahsoka. He was acting in his role as Supreme Chancellor of the Republic and his ruling was based on what they knew at the time.