We consider properties of long diffusive Josephson junctions where the weak link is a ferromagnet/normal metal bilayer with layers parallel to the current direction. In general such systems are governed by a two-dimensional Usadel equation which requires numerical solution. However, if the N layer is thin, the weak link can be adequately described by an effective one-dimensional Usadel equation with an effective exchange field h and an effective spin-flip rate Г. We derive this effective one-dimensional theory in various parametric regimes and calculate densities of states and supercurrent, comparing them to two-dimensional numerics and to experimental results. We find some peculiar results such as the non-monotonous dependence of the effective one-dimensional exchange field h on the real exchange field in the F layer.