Why is my city not generating more happiness after purchasing happiness structures?

Solution 1:

There are two kinds of happiness in Civilization V -> local happiness and global happiness.

Global happiness is the number displayed at the top of the screen. It is the sum of the local happiness of each city, plus any global happiness generated by other means (such as wonders, or luxury resources, as each unique resource generates global happiness).

Local happiness is the happiness generated by one specific city. This number can never increase past the size of the city (so a 6 population city can only generate 6 local happiness, regardless of how many happiness generating structures it builds).

Stadiums and Colosseums increase local happiness, which is why you did not see your happiness value change when they were built.

Some buildings (generally wonders) directly increase global happiness, without caring about the population size of the city they are built in. The Notre Dame wonder is an example of this.

Solution 2:

Happiness buildings cannot increase a city's happiness past it's population number. The effects of each happiness building stacks on the previous ones, so you would typically (as an example) need 2 population for each new happiness building you are creating. Otherwise it will do nothing.

This mechanic is implemented to prevent players from simply settling new cities late game and then using stored gold reserves to buy all of the happiness buildings at once, thereby boosting global happiness. The 1 population city would only be able to contribute +1 happiness until it's population grows.

Some buildings such as wonders ignore this rule and add directly to global happiness.