Your current simulation behaves pretty well, I would say. Because a rail wagon that is halfway a sharp bend, will indeed exhibit the behaviour that the wagon's center is no longer above the track, but next to the track.
Center of the tile
Traveling over the exact middle of a 32x32 tile: This is easily accomplished if you were to draw the rails ...
You can navigate to another URL by setting the variable window.location.href = url or with the function window.location.replace(url).
The difference between the two methods is how the browser's back-button will behave. The first one will generate a separate navigation history entry for each level while the second one will not.
But, you should consider if ...