We will only need 7 races to get the best three out of 25.

Let us Name horses as H1, H2, H3…H25.

Now,

Race 1 – H1 – H5
Race 2 – H6 – H10
Race 3 – H11 – H15
Race 4 – H16 – H20
Race 5 – H21 – H21

Let us now assume

First Rank of Race 1 – W11
Second Rank of Race 1 – W12
.
First Rank of Race 2 – W21
.
Third Rank of Race 2 – W23

Fifth Rank Of Race 5 – W55

Race 6 – W11 , W21 , W31, W41, W51 ( toppers from first five races)

Race 7 – 2nd from Race 6, 3rd from Race 6, X, Y , Z where

X= 2nd winner from the team of winner of race 6
Y= 3rd winner from the team of winner of race 6
Z= 2nd winner from the team of 2nd winner of race 6