Since both words have more than one meaning, it is difficult to give you a definitive answer without knowing the context. It would be helpful for you to use both words in a sentence. But I'll attempt to guess at an appropriate answer to your question based on the meaning that is most similar to both words. Common - done by many people. (http://learnersdictionary.com/definition/common) It is common practice for one town's fire department to help another town when there is a big fire. a common spelling mistake Normal - usual or ordinary : not strange (http://learnersdictionary.com/definition/normal) a normal day He had a normal childhood. These little setbacks are a normal part of life.

I think 'common' has to do with quantity and 'normal' has to do with quality. If I say suicide is common in a certain population, it would mean, it is in a large number in that population. If I say suicide is normal in certain population, it would mean it is socially acceptable in that population.

There's not much difference. They are almost interchangeable most of the time. "Common" simply means "frequent." "Normal" means "expected," It often carries the idea of "correct" and "proper." For example, in the United States, it is common to put ketchup on hamburgers and mustard on hot dogs. Some people put ketchup on hot dogs; that's less common, but it is normal. But, if someone put chocolate syrup on a hot dog, that would be very strange, and people would say "that's not normal."