The answer is 7 .Here’s how :
Divide the horses into group of 5 (A,B,C,D,E) and conduct 5 races to find 5 fastest horses .
Now conduct a race among these to find the fastest horse (Let’s say it’s from group E) .and let’s also assume that fastest horse from group D came 2nd and C came 3rd. Total 6 races done. Now the second fastest horse can be the one who came second in the group E which gave us the fastest horse or the one which came second in 6th race. Similarly the 3rd fastest horse can be
the either fastest horse from group D and C ,or the horses which came 2nd and 3rd in group E or the horse which came second in group D. Conduct a race among these 5 horses and you will get the result in final and 7th race.

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.

You have to assume the horses will not tire, which is not realistic. Any way, you would race horse # 1 with horse # 2 and the winner would race horse #3. The winner of that race then races horse #4. And so on down the line until horse # 25 races. Minimum # of races is 26.