The top diagram shows a 4-wheel cart has no delayed steering mechanism installed. Front wheels/Axle turns the same angle as shaft does.
The bottom diagram shows front wheel has delayed steering mechanism installed and how it works.
With a delayed steering mechanism. shaft and splinter bar are separated but are tied together at point S . Then a second or shaft turning circle center D in red is introduced. Adding a mechanical connection to hold shaft from S to D. It will control shaft's turning around only towards to circle center D. See left part of diagram. When an equine makes shaft to turn to its left, the shaft and splinter bar connecting point S moves away from original driving direction center line (in green) to left. S point on splinter bar is moved the same distance as S point on the shaft. Due to they are moving/turning toward to different radius centers, their turning angle margins are not totally synchronized. Though the S point on splinter bar and S point on shaft moved the same distance to left, to their own turning circle centers, each has turned a different angle. For instance the red shaft is turned 30 degrees to left, the splinter bar and front wheel/axle is turned about 25 degrees (reduced turning angle) This non-synchronized steering is so called Delayed Steering in carriage driving.
If move the shaft turning circle center further away from wheel/axle turning circle from D1 to D2 (in orange), the shaft even turned more margin (35 degrees) away from original direction line. the distance from D to S controls delayed steering margin between shaft and front wheels
The following diagram shows a single Marathon carriage that is equipped with a delayed steering mechanism. Equine and carriage has a different track path. Brown arrow line indicates equine's moving path. Green arrow line indicates cart moving path. Without delayed steering mechanism installed, the rare wheel of the cart may hit orange obstacles, this is the typical situation in high speed at sharp turn corners.