Modeling the propagation of hydraulic fractures is a complex problem that involves solving the equations that describe the intimate coupling between the cracked and porous solid phase and the fracturing fluid that drives the process according to established crack extension criteria. Most of the existing computational techniques become prohibitively expensive if applied to three-dimensional configurations, especially those involving multiple fractures because their accuracy relies on extremely fine discretizations of the geometry. These include the standard finite element method and standard displacement discontinuity method, which require substantial mesh densities in the vicinity of the crack front to capture the complex nature of the singular fields associated with fluid-driven cracks. Models have recently been proposed that offer sufficient accuracy and a significant increase in computational efficiency. These include the enhanced pseudo-3D model that explicitly incorporates a non-local elasticity relation and known crack-front asymptotic solutions that capture the multiscale near-front behavior. Here, we present an enhanced pseudo-3D model for simulating the propagation of multiple planar and nonplanar hydraulic fractures in a linear elastic medium. The algorithm is built on a fixed mesh with an adaptive tip element, and it incorporates leak-off, as well as stress interaction between the fractures. The model is validated against reference solutions of plane strain, radial, and multi-planar fracture geometries, for various sets of parameters that cover the majority of the problem parametric space.