"You turn left on this corner."
In this sentence, このかど and ここのかど are almost same. The difference is like "this corner" and "corner, here".
I think ”こっち” usually means "this way", whereas ”ここ” typically means "here". I'm not sure what determines whether it is "this way's corner" or "here's corner"; it seems to be that it should just be "このかど": "this corner".
曲がる is an intransitive verb so thinking of it like that might mislead you into thinking you can use 曲がる to mean "to turn something". 曲げる is the transitive form. を is also sometimes also used with intransitive verbs of motion to indicate a location passed through, the origin of the motion or the direction of an action.