Is a horse race purse size determined by the amount of betting on the race?

no the purse is determinded on the race when you bet you get some money fromo the purse if the horse u bet on wins

Actually, the Purse has nothing to do with the amount of money wagered in a horse race. The Handle is the total amount wagered on that race and after the track takes some out and the taxes are removed then the rest is divied up among the winning tickets. These are called Pools and there is one for each style/type of bet (Win Pool, Place Pool, Show Pool, Exacta Pool, Trifecta Pool, Pick 3 Pool, etc).