We present results from numerical simulations of nonlinear MHD dynamo action produced by three-dimensional flows that become turbulent for high values of the fluid Reynolds number. The magnitude of the forcing function driving the flow is allowed to evolve with time in such way as to maintain an approximately constant velocity amplitude (and average kinetic energy) when the flow becomes hydrodynamically unstable. It is found that the saturation level of the dynamo increases with the fluid Reynolds number (at constant magnetic Prandtl number), and that the average growth rate approaches an asymptotic value for high fluid Reynolds number. The generation and destruction of magnetic field is examined during the laminar and turbulent phase of the flow and it is found that in the neighborhood of strong magnetic flux ``cigars" Joule dissipation is balanced by the work done against the Lorentz force, while the steady increase of magnetic energy occurs mainly through work done in the weak part of the magnetic field.