I planted some rose of sharon starts from my mom's garden a couple years ago. They haven't produced any blooms yet. Also, it is kind of weird but the branches are shooting out very far and have really huge leaves, compared to the smaller leaf formations closer to the main body of the plant. Almost looks like two different plants in one. Should I prune these larger leafed stems off, and should I wait until next spring to do this?
Sounds like your Rose of Sharon is putting all of its energy into foliage instead of blooms. Pruning may help force blooms, but not this year.
You can prune the plant in late winter or early spring, while it's dormant.
This article will walk you through exactly how and when to prune your Rose of Sharon: