Intelligent reflecting surfaces (IRSs) are anticipated to provide reconfigurable propagation environment for next generation communication systems. In this paper, we investigate a downlink IRS-aided multi-carrier (MC) non-orthogonal multiple access (NOMA) system, where the IRS is deployed to especially assist the blocked users to establish communication with the base station (BS). To maximize the system sum rate under network quality-of-service (QoS), rate fairness and successive interference cancellation (SIC) constraints, we formulate a problem for joint optimization of IRS elements, sub-channel assignment and power allocation. The formulated problem is mixed non-convex. Therefore, a novel three stage algorithm is proposed for the optimization of IRS elements, sub-channel assignment and power allocation. First, the IRS elements are optimized using the bisection method based iterative algorithm. Then, the sub-channel assignment problem is solved using one-to-one stable matching algorithm. Finally, the power allocation problem is solved under the given sub-channel and optimal number of IRS elements using Lagrangian dual-decomposition method based on Lagrangian multipliers. Moreover, in an effort to demonstrate the low-complexity of the proposed resource allocation scheme, we provide the complexity analysis of the proposed algorithms. The simulated results illustrate the various factors that impact the optimal number of IRS elements and the superiority of the proposed resource allocation approach in terms of network sum rate and user fairness. Furthermore, we analyze the proposed approach against a new performance metric called computational efficiency (CE).