It's actually a very common problem among those of us who are very right-hand dominant; we use our right hands and arms for everything, and our left side may actually be a little underdeveloped.
I know that knowledge doesn't make it less frustrating, but all you can do is to be patient. Train both sides equally with the same weight and just let the left side catch up. When you do movement activities make sure you switch directions occasionally so that both sides get the same amount of training.
You can also do some extras throughout the day; get one of those squeezers to train the grip strength of your left hand periodically. Make an effort at home to use your left arm more. It's annoying and it takes patience, but your nondominant side will eventually catch up.