Which is the correct verb to use, "become" or "turn", when referring to one's age?

"Became" would be understood, but "turned" is the normally used verb, at least in the US. "Was" does not suit this context; it would more likely be used to reference the distant past; i.e., I was 20 when the Beatles arrived in America.

In my opinion turn would be an appropriate choice. Both may be used to denote changes. However, turn is used when a the process is still in continuation and gradual per se, like aging, while become is used when the phrase is more definitive and result oriented. For example:

• What do I have to do to become a pilot?
• He turned 100 years old yesterday.