1. Wrong: Their quickly swimming made them the better water polo team.
Better: Their quick swimming made them the better water polo team.
Explanation: Swimming is a gerund (noun) in this sentence, so the word describing it should be an adjective.
2. Wrong: Frankie reads so easy and fluent that nobody can believe he's only four.
Better: Frankie reads so easily and fluently that nobody can believe he's only four.
Explanation: Adverbs are needed to describe the verb reads.
3. Wrong: If the child behaves good, he can often earn more privileges.
Better: If the child behaves well, he can often earn more privileges.
Explanation: To describe the verb behaves, the adverb is needed.
4. Wrong: Ever since recovering from her horrible head cold, Harley smells good again.
Better: Ever since recovering from her horrible head cold, Harley smells well again.
Explanation: A head cold wouldn't make Harley not smell good -- a lack of hygiene or a serious workout could do that. That is, the verb smells (as with any of the sensory verbs) could be an action or linking verb. To describe how somebody does an action verb, use the adverb; to complete the idea of the linking verb, use the adjective.