LoRaWAN is a low-power wireless technology that provides long-range connectivity to battery-powered Internet of Things (IoT) devices. To minimize the energy consumption of the IoT nodes, LoRaWAN networks use for the uplink a pure non-slotted ALOHA multiple access scheme. Since the devices are not synchronized in time, collisions between uplink packets are the main source of errors when the number of nodes becomes important. To improve the reliability of dense LoRaWAN networks, we propose in this paper a successive interference cancellation LoRa receiver capable of decoding frames from two colliding users. The proposed two-user detector leverages the bit-interleaved coded modulation scheme of LoRa to improve the detection and cancellation of the strongest interfering user. We show that in the presence of two interfering users, the usage of a low coding-rate and iterative soft-detection are essential to attain error rates sufficiently close to the single-user scenario. Using network-level simulations, we subsequently evaluate the gains of our proposed two-user receiver in a realistic LoRaWAN network. To this end, we build advanced models of the studied receivers using Monte-Carlo simulations at the physical layer. For an overall packet error rate of 1%, simulation results indicate that a LoRaWAN network employing our two-user detector may serve 4.7 times more devices than a network with only a single-user receiver at the gateway.