Continuum robots are manipulators able to bend at any point of their structure. This feature allows continuum robots to avoid unwanted collisions in a complicated working area or grasp objects using their body. In this regard, they are used as instruments in industry and medicine. For continuum robots, a collision with an obstacle does not always mean that further motion is impossible. A part of continuum robot can be bent around the obstacle and pushed further. In connection with this point, we propose a planar motion planning algorithm for continuum robots considering the ability to bend over obstacles. We made two assumptions for the proposed algorithm. For the first assumption, friction is not affecting the robot shape while it is pushed forward. The second assumption suggests that a collision point divides a single section of the robot into two sections which can be described by the constant curvature approach. These assumptions allow us to present a single-section continuum robot going from one collision point to another as a multi-section continuum robot. The algorithm searches for the configurations going through collision points without crossing the obstacle borders. Obtained simulation results show that the proposed algorithm with the bending-over-obstacle feature copes well with the motion planning and is able to find multiple solutions less than 0.1 seconds.