Another way to think of this is tall vs. wide. To get a 20k victory, you need to treat one city like a diva, building the best culture buildings there. It's helpful to have the diva city *not* be your capital so that you can use the palace as a pre-build. As soon as a wonder finishes, start building the palace. Those shields can be transferred into the next wonder that the city wants to build. All the other cities are playing supporting roles. Making troops to conquer cities, to secure luxuries and resources. Making workers to ensure that all the tiles around the diva city are improved. Full disclosure: I've never achieved it, but I haven't tried very often.
I wonder if it is really the better idea to use anything less than the capital city to achieve 20000 culture in one city. My intuition tells me that in most cases the best approach would to start the game a one city challange. No settlers are built, instead the ancient age is used to build as many cheap wonders as possible. Only after the Statue of Zeus or Knights Templar have built you a power army you leave the path of the one city challenge. If no culture building is still availble to be build you may of course also built settlers. This approach of OCC at first, but pragmatism later could be fun.