Being unhappy because "its too crowded" is not related to the cultural border of the city. It is related soley to the population of that city. A size 12 city will have more unhappy people than a size 6 city, all things being equal.
So, how do you stop them from being unhappy??
1: Build a temple, colloseum and cathedral in the city. Each reduces the number of unhappy people.
2: Get luxuries, the more the merrier. Up to 8 luxuries can be enjoyed by the city, making people happy.
3: If you have over 2 luxuries, build a marketplace. It will increase the amount of "happy faces" your luxuries generate. See below.
4: Make some of your population specialists, particularly entertainers. This will help make your population happy/content.
5: As a final resort (IMO), increase your luxury spending just under the science spending slider.
Chart of "Happy Faces" produced by luxuries in a city with a marketplace.
1 luxury - 1 happy face
2 luxuries - 2 happy faces
3 luxuries - 4 happy faces
4 luxuries - 6 happy faces
5 luxuries - 9 happy faces
6 luxuries - 12 happy faces
7 luxuries - 16 happy faces
8 luxuries - 20 happy faces
Hope this helps
EDIT: Typo