In this paper, we propose a transmission control scheme for the approximate message passing (AMP)-based joint user identification and channel estimation in massive connectivity networks. In the proposed transmission control scheme, a transmission control function is designed to determine a user's transmission probability, when it has a transmission demand. By employing a step transmission control function for the proposed scheme, we derive the channel distribution experienced by the receiver to describe the effect of transmission control on the design of AMP algorithm. Based on that, we modify the AMP algorithm by designing a minimum mean squared error (MMSE) denoiser, to jointly identify the user activity and estimate their channels. We further derive the false alarm and missed detection probabilities to characterize the user identification performance of the proposed scheme. Closed-form expressions of the average packet delay and the network throughput are obtained. Furthermore, we optimize the transmission control function to maximize the network throughput. We demonstrate that the proposed scheme can significantly improve the user identification and channel estimation performance, reduce the average delay, and boost the throughput, compared to the conventional scheme without transmission control.