I assume you're trying to do all this with a single stroke?? Things really depend upon how you currently have objects constructed.
In order to get the appearance of an "overlap" you need an overlap. So draw a section of road to cover the overall path:
Same stroke style applied to an additional path on top of the base path. You won't be able to create the appearance of an overlap, regarding the white stroke, with a single path and the Appearance Panel.
To deal with those hairline bits that appear on the ends due to anti-aliasing, you can add an Effect the the base black stroke. This is really just a drawing anti-aliasing artifact and won't be visible on export (Art Optimized) or in print. But if it bothers you or you are uncertain it really isn't an issue, you can tweak things to remove it.
Highlight the base stroke... Choose
Effect > Distort & Transform > Transform from the menu. Enter a very small scale change in the top two fields of the Transform Options dialog window:
This will cause the black base stroke to be slightly larger than the underlying white stroke, and subsequently remove that hairline you may see.