Learn Polish: Jesteś coraz bardziej cierpliwy. - You are more and more patient.

In Polish, to express that a trait is progressively changing over time ("more and more" or "less and less"), you use the particle coraz followed by a comparative adjective or adverb. Translating "more and more" literally as bardziej i bardziej sounds unnatural to Polish ears.

Could I use a one-word comparative form of cierpliwy here?

Yes! The adjective cierpliwy has the one-word comparative form cierpliwszy. You could absolutely say Jesteś coraz cierpliwszy. However, using bardziej plus the basic adjective is very common in everyday speech, especially for longer words, and is perfectly correct.

If I am speaking to a woman, how does this sentence change?

You would change the masculine adjective cierpliwy to the feminine form cierpliwa, making it Jesteś coraz bardziej cierpliwa. The words coraz and bardziej do not change because they are not adjectives, so they have no gender.

Why is cierpliwy in the Nominative case? Doesn't the verb być (to be) usually take the Instrumental case?

When the verb być connects a subject directly to a lone adjective, that adjective stays in the Nominative case (the basic dictionary form). You only use the Instrumental case after być when the word following it is a noun, or a noun phrase like "a patient person" (cierpliwą osobą).