Quantitative yttrium-90 (Y-90) SPECT imaging is challenging due to the nature of Y-90, an almost pure beta emitter that is associated with a continuous spectrum of bremsstrahlung photons that have a relatively low yield. This paper proposes joint spectral reconstruction (JSR), a novel bremsstrahlung SPECT reconstruction method that uses multiple narrow acquisition windows with accurate multi-band forward modeling to cover a wide range of the energy spectrum. Theoretical analyses using Fisher information and Monte-Carlo (MC) simulation with a digital phantom show that the proposed JSR model with multiple acquisition windows has better performance in terms of covariance (precision) than previous methods using multi-band forward modeling with a single acquisition window, or using a single-band forward modeling with a single acquisition window. We also propose an energy-window subset (ES) algorithm for JSR to achieve fast empirical convergence and maximum-likelihood based initialization for all reconstruction methods to improve quantification accuracy in early iterations. For both MC simulation with a digital phantom and experimental study with a physical multi-sphere phantom, our proposed JSR-ES, a fast algorithm for JSR with ES, yielded higher recovery coefficients (RCs) on hot spheres over all iterations and sphere sizes than all the other evaluated methods, due to fast empirical convergence. In experimental study, for the smallest hot sphere (diameter 1.6cm), at the 20th iteration the increase in RCs with JSR-ES was 66 and 31% compared with single wide and narrow band forward models, respectively. JSR-ES also yielded lower residual count error (RCE) on a cold sphere over all iterations than other methods for MC simulation with known scatter, but led to greater RCE compared with single narrow band forward model at higher iterations for experimental study when using estimated scatter.