Solution to top three

The minimum number of races required is 7. The solution goes like this.
first divide the 25 horses into 5 grps A,B,C,D,E each containing 5 horses.
Now conduct 5 races for the 5 groups and u will get 1st,2nd,3rd place winners for each group.

Lemme call the 1st place winner of grp A as A1, 2nd place winner as A2 and so on till A5.
Do the same to the other groups.

Now group A1,B1,C1,D1,E1 together and conduct a race. let A1 be 1st, B1 be 2nd and C1 be 3rd.
A1 will be the fastest 1st place horse. Now for 2nd and 3rd place horse.

We r sure tat other horses in group C,D and E wont be the 2nd and 3rd place horses. C1 turned out to be the fastest horse in group C and so the other horses in C will be slower than C1.

So for the next race group A2,A3,B1,B2,C1 together and conduct the race.
The 2nd place winner in this race will be the 2nd fastest horse.
The 3rd place winner in this race will be the 3rd fastest horse.