Automatic crack segmentation plays an essential role in maintaining the structural health of buildings and infrastructure. Despite the success in fully supervised crack segmentation, the costly pixel-level annotation restricts its application, leading to increased exploration in weakly supervised crack segmentation (WSCS). However, WSCS methods inevitably bring in noisy pseudo-labels, which results in large fluctuations. To address this problem, we propose a novel confidence-aware co-training (CAC) framework for WSCS. This framework aims to iteratively refine pseudo-labels, facilitating the learning of a more robust segmentation model. Specifically, a co-training mechanism is designed and constructs two collaborative networks to learn uncertain crack pixels, from easy to hard. Moreover, the dynamic division strategy is designed to divide the pseudo-labels based on the crack confidence score. Among them, the high-confidence pseudo-labels are utilized to optimize the initialization parameters for the collaborative network, while low-confidence pseudo-labels enrich the diversity of crack samples. Extensive experiments conducted on the Crack500, DeepCrack, and CFD datasets demonstrate that the proposed CAC significantly outperforms other WSCS methods.