How many ways through the intersection?

Here is a "typical" intersection between main commuting roads in Bangkok:

Googlemaps link for this location: Ratchapruek and Boromarachachonani

I use quotes because it has a typical level of complexity, but almost every intersection is structured differently. Apparently, the civil engineers here wanted each interchange to be its own special little snowflake. One implication is that, for the drivers, it is very difficult to correctly navigate through the first time you encounter a new interchange. Also, if you are used to one transition through the interchange, you may be completely lost coming from a different direction or leaving a different direction.

Your mathematical challenge, should you accept it:
Count how many ways there are to go through this interchange

It might help to draw a diagram that abstracts the roads a bit, but makes it more clear. At least it will make it more clear to me what you've gotten wrong when I go to correct your work!