A downstream marching iterative scheme for the solution of the steady, incompressible, and two-dimensional parabolized or thin layer Navier-Stokes equations is described for a general curvilinear orthogonal coordinate system. Modifications of the primitive equation global relaxation sweep procedure result in an efficient marching scheme. This scheme takes full account of the reduced order of the approximate equations as it behaves like the SLOR method for a single elliptic equation. The proposed algorithm is essentially Reynolds number-independent and therefore can be applied to the solution of the incompressible Euler equations. A judicious choice of a staggered mesh enables second-order accuracy even in the marching direction. The improved smoothing properties permit the introduction of multigrid acceleration. The convergence rates are similar to those obtained by the multigrid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Numerical results are presented for several boundary-layer-type flow problems, including the flow over a spheroid at zero incidence.