We propose an iterative decoding method, which we call turbo-CS, for the reception of concatenated source-channel encoded sparse signals transmitted over an AWGN channel. The turbo-CS encoder applies 1-bit compressed sensing as a source encoder concatenated serially with a convolutional channel encoder. At the turbo-CS decoder, an iterative joint source-channel decoding method is proposed for signal reconstruction. We analyze, for the first time, the convergence of turbo-CS decoder by determining an EXIT chart of the constituent decoders. We modify the soft-outputs of the decoder to improve the signal reconstruction performance of turbo-CS decoder. For a fixed signal reconstruction performance RSNR of 10 dB, we achieve more than 5 dB of improvement in the channel SNR after 6 iterations of the turbo-CS. Alternatively, for a fixed SNR of -1 dB, we achieve a 10 dB improvement in RSNR.