This study presents and experimentally validates an adaptive control method for human-exoskeleton interaction through online adaptation of desired joint trajectories. Leveraging gait phase and human-exoskeleton interaction torque estimators, our approach enables seamless assistance adaptation to varying walking patterns and speeds. Specifically, a pre-trained neural network approximates the exoskeleton's dynamics, enabling real-time interaction torque estimation from kinematic measurements and commanded motor torques alone. These estimates drive a gradient-descent update of the joint reference trajectories, minimizing a cost function that penalizes both interaction torques and trajectory modification, ensuring bounded convergence and stability without user-specific parameter tuning. We compared our adaptive controller with a fixed-trajectory gait-phase-based controller during overground and treadmill walking at three self-selected speeds ranging from 0.4 to 0.8 m/s. In 16 participants, the adaptive controller significantly reduced the hip and knee interaction torques by 51.2%$\pm$11.1 and 63.9%$\pm$29.7, respectively, during overground walking. Muscular effort significantly decreased in Bicep Femoris (21.0%$\pm$34.5) and Rectus Femoris (28.1%$\pm$34.6), while remaining unchanged in other muscles. Cadence and gait speed increased by 7.6%$\pm$5.2 and 10.7%$\pm$8.3, respectively, indicating that participants could walk faster with less effort due to trajectory adaptation. Post-adaptation trajectories more closely resembled those of walking without the exoskeleton, and exoskeleton-torques aligned more closely with human biological torques. Our proposed adaptive controller, which requires only exoskeleton kinematics, also maintained performance during treadmill walking across speeds, demonstrating speed-invariant behaviour compared to the non-adaptive controller.