Continual learning has been a major problem in the deep learning community, where the main challenge is how to effectively learn a series of newly arriving tasks without forgetting the knowledge of previous tasks. Initiated by Learning without Forgetting (LwF), many of the existing works report that knowledge distillation is effective to preserve the previous knowledge, and hence they commonly use a soft label for the old task, namely a knowledge distillation (KD) loss, together with a class label for the new task, namely a cross entropy (CE) loss, to form a composite loss for a single neural network. However, this approach suffers from learning the knowledge by a CE loss as a KD loss often more strongly influences the objective function when they are in a competitive situation within a single network. This could be a critical problem particularly in a class incremental scenario, where the knowledge across tasks as well as within the new task, both of which can only be acquired by a CE loss, is essentially learned due to the existence of a unified classifier. In this paper, we propose a novel continual learning method, called Split-and-Bridge, which can successfully address the above problem by partially splitting a neural network into two partitions for training the new task separated from the old task and re-connecting them for learning the knowledge across tasks. In our thorough experimental analysis, our Split-and-Bridge method outperforms the state-of-the-art competitors in KD-based continual learning.