Humanoid robots have the potential to mimic human motions with high visual fidelity, yet translating these motions into practical, physical execution remains a significant challenge. Existing techniques in the graphics community often prioritize visual fidelity over physics-based feasibility, posing a significant challenge for deploying bipedal systems in practical applications. This paper addresses these issues through bounded residual reinforcement learning to produce physics-based high-quality motion imitation onto legged humanoid robots that enhance motion resemblance while successfully following the reference human trajectory. Our framework, Imitation to Control Humanoid Robots Through Bounded Residual Reinforcement Learning (I-CTRL), reformulates motion imitation as a constrained refinement over non-physics-based retargeted motions. I-CTRL excels in motion imitation with simple and unique rewards that generalize across five robots. Moreover, our framework introduces an automatic priority scheduler to manage large-scale motion datasets when efficiently training a unified RL policy across diverse motions. The proposed approach signifies a crucial step forward in advancing the control of bipedal robots, emphasizing the importance of aligning visual and physical realism for successful motion imitation.