We propose a learning-based Control Barrier Function (CBF) to reduce conservatism in collision avoidance for car-like robots. Traditional CBFs often use the Euclidean distance between robots' centers as a safety margin, which neglects their headings and approximates their geometries as circles. Although this simplification meets the smoothness and differentiability requirements of CBFs, it may result in overly conservative behavior in dense environments. We address this by designing a safety margin that considers both the robot's heading and actual shape, thereby enabling a more precise estimation of safe regions. Because this safety margin is non-differentiable, we approximate it with a neural network to ensure differentiability. In addition, we propose a notion of relative dynamics that makes the learning process tractable. In a case study, we establish the theoretical foundation for applying this notion to a nonlinear kinematic bicycle model. Numerical experiments in overtaking and bypassing scenarios show that our approach reduces conservatism (e.g., requiring 33.5% less lateral space for bypassing) without incurring significant extra computation time. Code: https://github.com/bassamlab/sigmarl