This paper studies intelligent reflecting surface (IRS) assisted full-duplex (FD) hybrid wireless powered communication networks (H-WPCN) to realize both reliability and self-sustainability functionalities. We propose an efficient periodic protocol where each battery-free device can harvest energy during an entire block except its own information transmission time and design a user-adaptive IRS beamforming scheme where different IRS phase shifts are adopted for uplink wireless information transmission (WIT). Then, a weighted sum rate (WSR) maximization problem of the joint design of the hybrid access point (HAP)/devices transmit power, time allocation, and IRS phase shifts by considering the practical imperfect self-interference cancellation (SIC) and the non-linear energy harvesting (EH) model is formulated. To solve the resulting non-convex optimization problem, an efficient algorithm based on the alternating optimization is proposed. Simulation results show that the user-adaptive IRS beamforming scheme significantly outperforms the static IRS beamforming scheme and also show the benefit of IRS for alleviating the practical hardware requirements for SIC.