Ditto BobRodes, but let me add one thing that may be the source of your confusion.
"Over its turn" is not a single idiom here. You're grouping the words wrong. "Not over" is one standard phrase, and "turn toward" is another.
To be "over" something as used here is to be finished with, to not be doing it or not care about it any more. Like, "I am over my drinking problem" means I don't have a drinking problem any more. "Sally is over Bob" means that Sally is not dating Bob any more or doesn't care about Bob any more. Etc. "Not over" means the opposite, the person still is doing it or caring about it.
"Turn toward" means, as Bob Rodes says, movement in a specified direction. Think of a person standing in a street or field who turns around to face in a different direction. When used literally, you might say something like, "I heard a noise to the left, so I TURNED TOWARD it to see what it was." Used metaphorically, like here, it means to consider or begin implementing an idea. Like, "When the bank wouldn't give me a loan, I turned toward relatives." Or, "As sales at home fell, our company turned toward the export market."