Convergence constrained power allocation (CCPA) in single carrier multiuser (MU) single-input multiple-output (SIMO) systems with turbo equalization is considered in this paper. In order to exploit full benefit of the iterative receiver, its convergence properties need to be considered also at the transmitter side. The proposed scheme can guarantee that the desired quality of service (QoS) is achieved after sufficient amount of iterations. We propose two different successive convex approximations for solving the non-convex power minimization problem subject to user specific QoS constraints. The results of extrinsic information transfer (EXIT) chart analysis demonstrate that the proposed CCPA scheme can achieve the design objective. Numerical results show that the proposed schemes can achieve superior performance in terms of power consumption as compared to linear receivers with and without precoding as well as to the iterative receiver without precoding.