It's wildly unintuitive, so don't feel bad. When I was first learning to ride a motorcycle with the MSF, I damn neared dumped the practice bike a few times trying to manhandle it. You'll really quickly internalize push-left-go-left. And it turns out to work great on regular bikes, too. (But it's terrifyingly more responsive - I actually can't relax pedaling a bicycle because of the fear of how easy it is to flip the things and capsize them from countersteering.)

EDIT: There's a neat heuristic that was brought up in the course: the center of mass of the bike is above the bottom of the wheels, and a turning bike must lean into the turn (weird road geometries excepted), therefore to turn the stuff under the center of mass must counter the top leaning in. Thus to go left, you push left, causing the front wheel to pull out from under the bike right-wards, which starts to capsize the bike leftwards! Like I said before, a bicycle reacts really really hard to this, but a hefty motorcyle has a far lower center of mass and leans into it more gracefully.