An improved apparatus including a localized heater that moves along the direction of conveyance with conveyed glass sheets to provide localized heating of a portion of each glass sheet to be bent without heating the rest of the glass sheet prior to its subsequent treatment. This invention is especially suitable for locally heating designated portions of glass sheets as they are heated while being conveyed through a furnace on a horizontal roller conveyor where the designated portion of glass sheets are to be formed with bends having relatively short radii of curvature at a bending station. The apparatus includes a pair of elongated localized gas heaters, each of which is positioned above the portion of the glass sheet which requires significant bending. These heaters are mounted on a carriage driven by a drive mechanism. The drive mechanism causes the carriage to move along a path parallel to the path of glass sheet conveyance to allow the heaters to continuously heat the designated portions of the glass sheet between an upstream position and a downstream position of the carriage. After the carriage reaches its downstream position, the drive mechanism causes the carriage to return to its upstream position in preparation for another cycle.