school-age child vs youth

school-age child and youth both are nouns.

Word	Noun	Adjective	Verb	Adverb
school-age child	Yes	No	No	No
youth	Yes	No	No	No

As nouns, youth is a hypernym of school-age child; that is, youth is a word with a broader meaning than school-age child:

school-age child: a young person attending school (up through senior high school)
youth: a young person (especially a young man or boy)

Other hypernyms of school-age child include spring chicken, young person, younker.

school-age child (noun)	youth (noun)
a young person attending school (up through senior high school)	the freshness and vitality characteristic of a young person
	young people collectively
	a young person (especially a young man or boy)
	early maturity; the state of being young or immature or inexperienced
	the time of life between childhood and maturity
	an early period of development