When Forrest and Jenny were running in the field to the massive tree where they sat on a branch, Forrest was running properly and bending his legs. But, when the bullies were chasing after him on bikes down the dirt path, he wasn't bending his legs and running like he did in the field. How come?
One very reasonable explanation is that Forrest outgrew the braces. Based on their construction, this would start misaligning the knee and brace joint, making it hard to bend your legs while in the braces.
This is no different from how it's really difficult to keep child prosthetics appropriately sized at all times, given their cost to replace and how fast children grow.
Or, more boringly, maybe his braces started showing wear and tear in the joints.
We also don't know the details of Forrest's condition. Maybe his muscles are stiffer after a period of no exercise, and he had been running for a while before we see him run with Jenny; we just didn't see the initial stiffness in his legs.
All in all, there isn't any explicit proof mentioned in the movie as to why these two scenes are different, but there are plausible explanations.
Forrest's braces were equipped with droplocks at the knee joints. To bend your knees (in order to be able to sit down) you had to lift the locks with your hands and keep your knees sufficiently bent. When you straightened your legs again the droplocks automatically dropped over the knee joints and blocked the joints.
So if Forrest took care not to straighten his legs up to a certain degree he could keep them sometimes bent (which was of course not the intention, the purpose of the droplocks was to prevent you from bending your knees). The moment he straightened his legs the droplocks would activate and he could then only walk with straight legs. When he had to flee quickly from the bullies he simply did not have the frame of mind in order to 'cheat' the braces, a thing he did when he danced and when he ran in the field near the tree)
In those days up until the 1960s and 1970s children got equipped with very elaborate braces for far more orthopedic conditions than in more recent times. I recognised the lateral circular knee pads on Forrest's braces that were used in the case of braces for genu valgum (knocked knees) to push the knees outwards.
If you had in those days a persistent case of knocked knees and flat feet you would probably end up with those kinds of braces with high boots and droplocks around the age of 8 (if you were lucky you had to wear them only at night but in more persistent cases during daytime)
This would explain why Forrest had 'strong legs' and could walk and even run without the braces. And why he got braces to avoid back problems (due to the bad positioning of his feet and knees).