Owing to the inherent central information processing and resource management ability, the is a promising network structure for an intelligent and simplified sixth-generation (6G) wireless network. Nevertheless, to further enhance the capacity and coverage, more radio remote heads (RRHs) as well as high-fidelity and low-latency fronthaul links are required, which may lead to high implementation cost. To address this issue, we propose to exploit the as an alternative way to enhance the C-RAN, which is a low-cost and energy-efficient option. Specifically, we consider the uplink transmission where multi-antenna users communicate with the baseband unit (BBU) pool through multi-antenna RRHs and multiple IRSs are deployed between the users and RRHs. RRHs can conduct either point-to-point (P2P) compression or Wyner-Ziv coding to compress the received signals, which are then forwarded to the BBU pool through fronthaul links. We investigate the joint design and optimization of user transmit beamformers, IRS passive beamformers, and noise covariance matrices to maximize the uplink sum rate subject to fronthaul capacity constraints under P2P compression and Wyner-Ziv coding. By exploiting the Arimoto-Blahut algorithm and semi-definite relaxation (SDR), we propose a successive convex approximation approach to solve non-convex problems, and two iterative algorithms corresponding to P2P compression and Wyner-Ziv coding are provided. Numerical results verify the performance gain brought about by deploying IRS in C-RAN and the superiority of the proposed joint design.