You're not "losing" culture, you need to have a certain amount of culture before you can improve the city. That culture doesn't actually go away, you still have a total amount of culture that is equal to what you had before upgrading, you just don't have as much excess culture, which is what gives the bonus.
Think of it this way: hunter/gather societies didn't really work up their "culture", it wasn't until populations took root and started to create what we think of as "culture" that actual cities began to grow. All across history, the civilizations that increase their culture tend to thrive, but that culture is what creates some of the world's greatest innovations. You're not really "using up" culture, but you need to reach a certain amount of culture to be able to understand the innovations needed to upgrade. Anything extra is just a bonus - exactly as it is in the game.