Do a search on google for "counter steering". The vast majority of the hits will be motorcycle sites but motorcycles are bicycles with engines. They both steer the same.
You gotta bend your mind around the idea that two wheel single track vehicles (bicycles and motorcycles) steer totally differently from 4 wheeled dual track vehicles (cars and trucks). With counter steering there's three parts to any turn.
1) The tip in where you induce your bike to "fall" in the direction of the turn by steering away from the turn for a split second or with a very slight pressure on the bars. You'll see this on the motorcycle sites as "push right to go right". When you push on the right grip you angle the front wheel to the left which makes the bike start to fall to the right.
2) Catch and balance. This is where you catch and correct the initial fall into the turn by turning the bars into the turn and re-establish your balance but at the lean angle needed for the turn. In practice this is done smoothly and seamlessly following the initial turn away of the bars that starts the tip in.
3) Recover to upright or switch direction. The final part of the turn is usually a return to upright. This is where you turn the bars into the turn harder to bring the bike back in and under yourself. If you hold the angle a little longer you'll pass through the upright position and drop to the other side and enter a new turn in the other direction.
If you're more used to driving a car you may well be conciously trying to steer it "normally" rather than steering it using this counter steering method. Cars and bicycles have absolutely nothing at all in common when it comes to steering. If you are confusing the two methods your brain and muscle signals will be fighting each other and wobbles is the outcome. You sort of make it around the turn or obstacle but when it's all over you're not really sure how, right?
You should go out and work on learning and using counter steering conciously. Ride along an open road in a straight line and lightly push on one hand grip. Note how the bike falls into a turn to that side. Push on the opposite grip to make the bike stop dropping into the turn or even a little more and make it come back up or go the other way. Use small but conciously applied pressures when you're doing this. Light pressure makes it tip into the slowly. More pressure makes the turn in snappy.
When you do it this way you'll find that the bike responds instantly and very accurately to your slightest bar pressure inputs. Keep working on using this conciously until it's second nature.
A big open and empty shopping center parking lot is a great place to work on this. When you get the hang of steering this way and doing it in a fully concious manner and under your complete control you can actually toss out some pennies ahead of you in a random pattern and then steer accurately enough to ride your front wheel over them even though you end up turning back and forth. And you can do it under complete and comfortable control. Guaranteed no more wobbles unless you're riding back from the pub....
