You are asking the basic question of forming NP (noun phrase) in English. Actually, there may be more than two ways to for NP, yet I won't go that far. 1. NP--> adj+N. the focus here is the latter noun, and the adj. before it MODIFIES the feature/character of the N. 2. NP--> N1+N2 the focus is for N2, and the N1 identifies the origin/source of the N2. e.g. silk shirt, book store. Specially, for this case, there mostly are "settled" phrases, like "hot dog" (as the fast food) v.s. "hot dog" (as a dog that is hot). 3.NP--> N1 + of N2 the focus is the N1, and the of N2 signifies the possessor of N1. And this is for nouns of non-living things, like the map of the World, the wall of the classroom, the virtue of honesty. Specifically, when the nouns are living things, we use apostrophe (') to indicate possessor, like The horse's tail, Jim's sister or metaphorically, today's news. Above are the linguistic reasons for your questions if you do want to know why and how thoroughly. Yet it is as simple to fit the rules next time and as the more you learn, the better you understand.

Rantao's answer is very good. It may be hard to apply the theory in practice, and so it's important to find authentic examples. I recommend the sentence database fraze.it I searched for "race" + "horse". It's worth checking a few pages as not all results will be relevant. You can then start to identify common patterns. You will notice that no sentences include "horse of race", however many include "racehorse". The more you do this, the more the theory will make sense to you.

If people just do not say it that way, why bother which one is correct. Language comes before grammar. I do not know who said similar thing like this. Hope this helps.