Those of you who were not cutting math classes to go to Aqueduct when you were younger, might remember the formula 2*pi*r (replace "pi" with the Greek symbol that I cannot reproduce in this post), which is the formula for the circumference (distance around the perimeter) of a circle. Pi is a constant equal to about 3 so the distance around a circle is about 6 times the radius ("r" stands for radius of the circle). Also note that the radius is the distance from the center to the outer edge of the circle.

If you think of the turn on a track as a semi-circle then the distance you travel along the rail in the turn would equal about 3r. the distance you would travel 1 path out from the rail would be an extra 3*(the width of a path). It does not matter what "r" is because the delta in distance is the same regardless.

However, if you have a tight turn (smaller "r"), the outside horse has to travel faster to keep pace with the rail horse because it has to make up that extra distance around the turn over a shorter total distance of ground (because the total distance around the turn is less that it would be on a more sweeping turn with a bigger "r").

It might help to picture a merry-go-round with a radius of about 10 ft. Very near the center (say 1 ft) you are spinning slowly because you have very little distance to travel during each revolution (think of this as the rail horse). Halfway between the center and the outside edge you are spinning faster (the 2 path) in order to keep up with the person near the center because you have to travel farther during each revolution. On the outside edge you are spinning even faster (the 3 path) to keep up with the other 2 because you are traveling even farther still during each revolution. On a small merry-go-round (r=10ft) you would be spinning much faster on the outside edge than someone 5 feet inside of you (twice as fast). On a massive merry-go-around (r=50ft), the difference in speed if you were on the outside edge and someone was 5 feet inside of you would not be nearly as great (not anywhere near twice as fast). However, in both cases you would be traveling the same "extra distance" of about 30 feet during each revolution.

By now your head might be spinning. The bottom line is that ground loss per path is the same regardless of how tight the turns are but tight turns and banking do affect how much faster the outside horses have to travel (and how much harder they have to work) to keep up with the inside horses.

There is a lot of physics that also complicates this but I will not venture down that path.

Chris