We present a novel, highly scalable and optimized solver for turbulent flows based on high-order discontinuous Galerkin discretizations of the incompressible Navier-Stokes equations aimed to minimize time-to-solution. The solver uses explicit-implicit time integration with variable step size. The central algorithmic component is the matrix-free evaluation of discretized finite element operators. The node-level performance is optimized by sum-factorization kernels for tensor-product elements with unique algorithmic choices that reduce the number of arithmetic operations, improve cache usage, and vectorize the arithmetic work across elements and faces. These ingredients are integrated into a framework scalable to the massive parallelism of supercomputers by the use of optimal-complexity linear solvers, such as mixed-precision, hybrid geometric-polynomial-algebraic multigrid solvers for the pressure Poisson problem. The application problem under consideration are fluid dynamical simulations of the human respiratory system under mechanical ventilation conditions, using unstructured/structured adaptively refined meshes for geometrically complex domains typical of biomedical engineering.