Learn Swedish: Han fyller nitton år i maj.

The verb fyller literally means 'fills'. In Swedish, when talking about age, you do not 'turn' or 'become' an age; you 'fill' that number of years. The general expression for having a birthday is att fylla år.

Since the current lesson is about dates, how would I change this to say 'He turns nineteen on the fifth of May'?

You would use the exact date format from this lesson: Han fyller nitton år den femte maj. Notice that you add den before the ordinal number (femte) and you remove the preposition i when giving the specific date.

Why is maj written with a lowercase 'm'?

Unlike in English, Swedish does not capitalize months, days of the week, or languages unless they are the very first word in a sentence.

Why doesn't the word for 'years' (år) change to a plural form after nitton?

Because ett år is an ett-word (neuter noun) that ends in a consonant. In Swedish, these types of nouns have the exact same form in the singular and the indefinite plural. So 'one year' is ett år, and 'nineteen years' is nitton år.

Could I leave out år and just say Han fyller nitton i maj?