This paper explores the use of an intelligent reflecting surface (IRS) in backscatter-assisted wireless powered non-orthogonal multiple access (NOMA) networks. In these networks, energy-constrained sensors first harvest energy from a power station with the help of an IRS in the downlink (DL) phase. They then transmit information to an access point using NOMA during the uplink (UL) phase. When sensors have low energy levels, they can switch from the high-power conventional mode to the low-power backscatter mode. By utilizing a practical energy harvesting model that captures realistic rectifier characteristics, we propose a new algorithm that optimizes DL energy harvesting time, UL backscatter time, UL conventional data transmission time, and DL phase-shift vector. Additionally, we analyze the impact of imperfect channel estimation on system performance. The numerical results demonstrate that IRS provides at least 30 % energy efficiency improvements.