Implementing obstacle avoidance in dynamic environments is a challenging problem for robots. Model predictive control (MPC) is a popular strategy for dealing with this type of problem, and recent work mainly uses control barrier function (CBF) as hard constraints to ensure that the system state remains in the safe set. However, in crowded scenarios, effective solutions may not be obtained due to infeasibility problems, resulting in degraded controller performance. We propose a new MPC framework that integrates CBF to tackle the issue of obstacle avoidance in dynamic environments, in which the infeasibility problem induced by hard constraints operating over the whole prediction horizon is solved by softening the constraints and introducing exact penalty, prompting the robot to actively seek out new paths. At the same time, generalized CBF is extended as a single-step safety constraint of the controller to enhance the safety of the robot during navigation. The efficacy of the proposed method is first shown through simulation experiments, in which a double-integrator system and a unicycle system are employed, and the proposed method outperforms other controllers in terms of safety, feasibility, and navigation efficiency. Furthermore, real-world experiment on an MR1000 robot is implemented to demonstrate the effectiveness of the proposed method.