The sequence that this question is asking about can be found in chapter 12, which is titled "A Stormy Day." Beauty is the horse that has been charged with taking Gordon and John on a somewhat lengthy trip, and the weather is not good. On the way out, the men notice that the river is already swollen with the rain. By the time the men are headed home, half of the bridge isn't even visible. The water is washing over it, but the men decide to coax Beauty across it.
Beauty refuses to go across the bridge, despite a couple of flicks of the whip. It turns out that a section of the bridge has washed away, and crossing the bridge would have likely seen horse and men pulled into the water and drowned.
Beauty's refusal to go across the bridge isn't necessarily a major turning point in the story. It functions as another rising action that further shows readers and characters that Beauty is a good, dependable horse with good sense. You could argue that the refusal changed the story by allowing Beauty to survive. If Beauty had died, the book would have ended.
A better argument might be to discuss how Beauty's refusal saved the men. Squire Gordon ends up getting rid of Beauty, but that happens after men like John are allowed to consistently treat Beauty well. Had John and Gordon died, Beauty most likely never would have met and been put under the care of Joe Green. While Joe made some early mistakes, Joe turns out to be one of the best of Beauty's caretakers, and we rejoice deeply when Beauty is reunited with Joe at the end of the story. None of that would have likely happened had Beauty crossed the bridge and killed John and Gordon.