John Paul Jones moved to America to avoid a trial that involved his handling of a mutiny.

simple sentence
compound sentence
complex sentence (my answer)
compound complex sentence

Correct.

John Paul Jones moved to America to avoid a trial that involved his handling of a mutiny.

John Paul Jones moved to America to avoid a trial that involved his handling of a mutiny.

Explanation:
This sentence is a complex sentence. It consists of two clauses, an independent clause ("John Paul Jones moved to America") and a dependent clause ("to avoid a trial that involved his handling of a mutiny"). The dependent clause cannot stand alone as a complete sentence because it relies on the independent clause to give it meaning.