The way I build barrel tunrs is that I teach them to move off your leg, first. Then I'll start by ten feet circles around the barrel and slowly decrease the size. Then, when you get to the big time tight circles add the vocal cue of "Hey hey!" or whatever else and pull her into a tight turn. And when they dive in and turn, reward them. My horses turns have improved dramatically, even though that's really simple and I don't know if that's the "Official" way to do it. I just know it works for me and I ahven't ruined a horse yet so it must be good enough, huh?
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.