Flexible polymers crystallize with chain folding, which shows a unique phenomenon called self-poisoning. As a result, minima in crystal growth rates of strictly monodisperse short-chain polymers are observed near the temperatures of transitions from extended-chain to once-folded-chain growth, from once-folded to twice-folded growth, etc. We employed dynamic Monte Carlo simulation of lattice polymers to visualize such transitions in molecular details. We observed a rate crossover between chain extension and lateral growth of polymer lamellar crystals at the wedge-shaped growth front, resulting in a rate minimum around the melting point of the metastable once-folded lamellar crystal. The rate minimum can be interpreted as due to the dependence of crystal growth rates on the excess crystal thickness beyond the minimum stable thickness. Furthermore, during crystal thickening, numerous molten chains are shown to be sucked into the lamellar crystals through the basal planes, demonstrating an important source of crystallinity from secondary crystallization lagging behind the crystal growth front.