race vs horse-race

race is a noun but horse-race is not a noun.

Word	Noun	Adjective	Verb	Adverb
race	Yes	No	Yes	No
horse-race	No	No	Yes	No

As verbs, horse-race is a hyponym of race; that is, horse-race is a word with a more specific, narrower meaning than race:

Other hyponyms of race include show, place, boat-race, campaign, run, speed skate.

race (noun)	horse-race (noun)
a canal for a current of water
a contest of speed
any competition
people who are believed to belong to the same genetic stock
(biology) a taxonomic group that is a division of a species; usually arises as a consequence of geographical isolation within a species
the flow of air that is driven backwards by an aircraft propeller

race (verb)	horse-race (verb)
compete in a race	compete in a horse race
move hurridly
cause to move fast or to rush or race
to work as fast as possible towards a goal, sometimes in competition with others