It's funny because I have the exact opposite problem you do. My bike dies when the handle bar is turn all the way to the right. It operates normally but when the handle bar is turned to the right I can't start the bike. If the bike is running and I turn handle bar to the right then it dies. It actually dies just a little bit before the very end of the turn.
I've been lazy and haven't attempt diagnosis/repair, but it looks like when it's turn to the right, it tugs on the wires that are under the left gas tank (neck) cover. Since your issue happens on left turn, then you might want to check the wires under the right gas tank neck cover. I believe that is the starter button wire so it's definitely a point of interest. Maybe follow that wire from end to end to see if there is any abnormal pressure on it or any damages. Also check for any loose wire/connection or abnormalities in the headlight housing.