Flapping-wing micro aerial vehicles offer quieter and safer operation than rotary-wing drones, yet achieving precise autonomous control of bird-scale ornithopters remains challenging: lift, airspeed, and turning authority are tightly coupled and governed by only a few control inputs. Conventional cascaded controllers treat altitude, speed, and heading independently, producing persistent tracking errors during complex maneuvers, while time-parameterized trajectory tracking requires predefined speed profiles that existing methods cannot robustly produce for these coupled dynamics. We address both limitations simultaneously with a Model Predictive Contouring Control (MPCC) approach that tracks arc-length-parameterized trajectories while optimizing progress online, eliminating the need for predefined timing. However, MPCC requires a dynamical model that captures the coupled aerodynamics without exceeding the computational budget of real-time nonlinear optimization. Here, we propose a compact, continuously differentiable model that captures the dominant couplings of bird-scale ornithopters, enabling real-time predictive control. We validated the method with the XFly ornithopter flying along circular and three-dimensional racing trajectories and achieved a mean deviation from the reference trajectory between 6.5 and 9 cm at speeds up to 3 m/s, which represents an almost 10-fold improvement over prior ornithopter control methods.