Is it normal, for a once broken limb to still be weak after it healed?

When I was about 5 or 6 years old, I fractured my elbow on the swing set that was across from my kindergarten. I had been playing with a friend and both of our moms were pushing us on the swings. At some point, I got the strange idea in my head to let go of the chains and see if I would still be okay. I did it successfully for about two times, but on the third time I failed. I remember flying off of the seat and being happy because I felt like I could fly, but then I somehow took a u-turn and crashed into the wood chips on the playground. I was crying and in so much pain and about 30 mins or an hour later my mom took me to get X-rays. I'm not sure if I felt weak while my fractured elbow was mending itself, but I do know that after the cast was taken off my arm it felt "weak". The weakness never stopped. I'm in my late teens now and I only recently realized that this whole time the function in my left arm was behind the function of my right. Every since the incident, my right arm/hand were always stronger than the left. I can do so many things with my right that I can't of with my left. I can't eat with utensils and I can't even properly use a pencil with my left hand. My right hand is the dominate one and left is a submissive! I can only use it for supporting activities. I hope this isn't confusing to anyone, but I just want to know if this is normal. Has anyone had this happen to them?