hair style vs hairstyle

hair style and hairstyle both are nouns.

Word	Noun	Adjective	Verb	Adverb
hair style	Yes	No	No	No
hairstyle	Yes	No	No	No

As nouns, hair style and hairstyle are synonyms defined as:

hair style and hairstyle: the arrangement of the hair (especially a woman's hair)

Other synonyms of hair style include coif, coiffure, hairdo.

hair style (noun)	hairstyle (noun)
the arrangement of the hair (especially a woman's hair)	the arrangement of the hair (especially a woman's hair)