The strength-ductility tradeoff has been a common long-standing dilemma in materials science. For example, superplasticity with a tradeoff in strength has been reported for Cu50Zr50 nanoglass (NG) with grain sizes below 5 nm. Here we report an improvement in strength without sacrificing superplasticity in Cu50Zr50 NG by using a bimodal grain size distribution. Our results reveal that large grains impart high strength, which is in striking contrast to the physical origin of the improvement in strength reported in the traditional nanostructured metals/alloys. Furthermore, the mechanical properties of NG with a bimodal nanostructure depend critically upon the fraction of large grains. By increasing the fraction of the large grains, a transition from superplastic flow to failure by shear banding is clearly observed. We expect that these results will be useful in the development of a novel strong and superplastic NG.