We reveal the existence of systematic variations of isobaric fragility in different supercooled Lennard-Jones binary mixtures by performing molecular dynamics simulations. The connection between fragility and local structures in the bulk is analyzed by means of a Voronoi construction. We find that clusters of particles belonging to locally preferred structures form slow, long-lived domains, whose spatial extension increases by decreasing temperature. As a general rule, a more rapid growth, upon supercooling, of such domains is associated to a more pronounced super-Arrhenius behavior, hence to a larger fragility.