Still left trying to visualise how directing rider weight forward into a turn while pushing wheel in opposite direction works.
The easiest way to visualize what is going on is to take a small broom, place the top of the handle in your palm and balance the broom. After you have danced around for a while trying to keep the handle underneath the wider end of the broom you may see a pattern. When the straw part of the broom is balancing, if you move the handle to your left, the broom falls to the right, and vice versa.
If you think of the palm of your hand as the front tire contact point on the pavement, when you move that contact point to the left the bike goes to the right. There are all sorts of other things going on, but essentially, when you are turning the handlebars you are putting the tire contact patch in a place where it will lean the bike left, or right or hold it straight.
The faster you ride the bike the more sensitive it becomes to the tire contact patch movement. If you are on a dirt bike going slowly we all have seen the very quick left and right turning of the fork that is necessary to stay balanced at a low forward speed. At slow speeds, you have to overcorrect continually to stay balanced and maintain the line you wish to take. This is what you are doing with the broom, mostly.
David