Contact-rich manipulation tasks often exhibit a large sim-to-real gap. For instance, industrial assembly tasks frequently involve tight insertions where the clearance is less than 0.1 mm and can even be negative when dealing with a deformable receptacle. This narrow clearance leads to complex contact dynamics that are difficult to model accurately in simulation, making it challenging to transfer simulation-learned policies to real-world robots. In this paper, we propose a novel framework for robustly learning manipulation skills for real-world tasks using simulated data only. Our framework consists of two main components: the "Force Planner" and the "Gain Tuner". The Force Planner plans both the robot motion and desired contact force, while the Gain Tuner dynamically adjusts the compliance control gains to track the desired contact force during task execution. The key insight is that by dynamically adjusting the robot's compliance control gains during task execution, we can modulate contact force in the new environment, thereby generating trajectories similar to those trained in simulation and narrowing the sim-to-real gap. Experimental results show that our method, trained in simulation on a generic square peg-and-hole task, can generalize to a variety of real-world insertion tasks involving narrow and negative clearances, all without requiring any fine-tuning. Videos are available at https://dynamic-compliance.github.io.