You say you don't lean far enough? You lean just the right amount, everytime, otherwise you'd fall off to the outside of the turn
What you mean is you don't lean as far as some others do. Lean can be varied in a turn, what you're doing is maintaining the centre of gravity at an angle in a turn. My bike leans more than others because I keep my body upright. When race bikers lean off the bike it's actually to keep the bike more upright so they don't ground the pegs, but the centre of gravity remains the same whether you lean your body with or against the bike, and the bike will lean less or more respectively.
And trust me, at around a jogging pace, you're likely to be counter steering. If you do slow manoeuvres like the slalom on the CBT or the figure 8, you DO steer normally because you are doing it at a walking pace, and you notice that the bars are moving more, but the bike is less stable as you don't have much gyroscopic forces helping stabilise the bike. In a counter steer you do it to upset the gyroscopic forces, it's a kind a gyroscopic precession (yawn) that you don't need to know about really, but knowing push left to go left, and to go left some more, you push some more is important. Your body will compensate subconsciously usually. When you're up and running, go find an empty car park and ride at 10mph or thereabouts and gently push the bars one way or another. When you have a turn started and space to continue the turn, push again (gently) on the bar in the direction of the turn and see what happens.
So if you can push left to go left, next (gently) try pulling right to go left, it will reinforce the theory.