This paper investigates intelligent reflecting surface (IRS)-aided multi-antenna wireless powered communications in a multi-link interference channel, where multiple IRSs are deployed to enhance the downlink/uplink communications between each pair of hybrid access point (HAP) and wireless device. Our objective is to maximize the system sum throughput by optimizing the allocation of communication resources. To attain this objective and meanwhile balance the performance-cost tradeoff, we propose three transmission schemes: the IRS-aided asynchronous (Asy) scheme, the IRS-aided time-division multiple access (TDMA) scheme, and the IRS-aided synchronous (Syn) scheme. For the resulting three non-convex design problems, we propose a general algorithmic framework capable of suboptimally addressing all of them. Numerical results show that our proposed IRS-aided schemes noticeably surpass their counterparts without IRSs in both system sum throughput and total transmission energy consumption at the HAPs. Moreover, although the IRS-aided Asy scheme consistently achieves the highest sum throughput, the IRS-aided TDMA scheme is more appealing in scenarios with substantial cross-link interference and limited IRS elements, while the IRS-aided Syn scheme is preferable in low cross-link interference scenarios.