Intelligent reflecting surface (IRS) is envisioned to play a significant role in future wireless communication systems thanks to its powerful capability of enabling smart and reconfigurable radio environment. In this paper, we study the multi-IRS aided downlink communication in a massive multiple-input multiple-output (MIMO) system, where a multi-antenna BS simultaneously serves multiple remote single-antenna users with orthogonal beams reflected by multiple IRSs. By exploiting the line-of-sight (LoS) link between each pair of selected IRSs, a multi-hop cascaded LoS link can be established between the BS and each user via their cooperative beam routing. Under this setup, we optimize the selected IRSs and their beam routing path for each user, along with the BS/IRS active/passive beamforming such that the minimum received signal power among all users is maximized, subject to a new multi-beam routing path separation constraint for avoiding the inter-user/route interference. To tackle this problem, we first derive the optimal BS/IRS active/passive beamforming in closed-form for any given beam routes and show the beam routing optimization is NP-complete by recasting it as an equivalent graph-optimization problem. To solve this challenging problem, we then propose an efficient recursive algorithm to partially enumerate the feasible solutions, which effectively balances the performance-complexity trade-off by tuning its design parameter. Numerical results demonstrate that the proposed algorithm can achieve near-optimal performance with low enumeration complexity and also outperform other benchmark schemes.