My answer is 11.

First you split the 25 horses into 5 groups of 5. The puzzle tells us that we can’t measure speed, only relative speed, so we can’t compare horses across races. Thus, we have to take the top 3 horses of each race.  (5 races)

Next, we have 15 horses so we split them into 3 groups of 5. We again take the top 3 of each group.  (3 races)
Then, from the remaining 9, we form one group of 5 and take the top 3, adding it to the remaining 4. (1 race)
Now we have 7 remaining, we form another group of 5 and take the top 3, adding it to the remaining 2. (1 race)
We now have 5 horses left and we can form our top 3 in this final race. (1 race)

Add them up and you’ve got 11 races.