Regardless of growth plates, exercise does not cause bones to get longer or wider than they would have without exercise.
Each bone is like a sponge, it grows to a predetermined size as regulated by your DNA. Exercise makes the bone more dense and stronger. The more you exercise, the more stress your bones are under. The bones respond by getting denser and stronger.
Like most organs/tissues, skeletal bone is constantly remodeling. Bone remodeling is a lifelong process where mature bone is removed (a process called resorption) and new bone is formed (called ossification). Cells called osteoclasts are responsible for the breakdown (resorption) while cells called osteoblasts handle new bone ossification. When we are young, we make new bone faster than we remove old bone and we increase our bone mass. Through our 20s this starts to shift, and we begin to lose bone mass faster than it is created. Source