An engine is provided with a cylinder head having a bridge region surrounded by an exhaust face, an exhaust passage intersecting the exhaust face, and an exhaust gas recirculation (EGR) passage fluidly coupled to the exhaust passage and intersecting the exhaust face. The head defines an upper cooling jacket having a cavity or fluid passage extending from the jacket towards a head deck face and to a closed end wall within the bridge region. The cylinder head is cooled by directing coolant from a lower jacket to an upper jacket via a drill passage adjacent to an exhaust face of the head, diverting the coolant exiting the drill passage into the fluid passage or cavity along a rib. Coolant is then directed from the fluid passage into an EGR cooling passage formed by the upper jacket adjacent to the exhaust face and about the EGR passage.