Reliable communication over bandlimited and non-linear channels usually requires equalization to simplify receiver processing. Equalizers that perform joint detection and decoding (JDD) achieve the highest information rates but are often too complex to implement. To address this challenge, model-based neural network (NN) equalizers that perform successive interference cancellation (SIC) are shown to approach JDD information rates for bandlimited channels with a memoryless nonlinearity and additive white Gaussian noise. The NNs are chosen to have a periodically time-varying and recurrent structure that imitates the forward-backward algorithm (FBA) in every SIC stage. Simulations for short-haul fiber-optic links with square-law detection show that NN-SIC nearly doubles current spectral efficiencies, and bipolar or complex-valued modulations achieve energy gains of up to 3dB compared to state-of-the-art intensity modulation. Moreover, NN-SIC is considerably less complex than equalizers that perform JDD, mismatched FBA processing, and Gibbs sampling.