A golf club head comprising a crown with an edge support structure and a flexible, conformal weight sized to fit within the edge support structure is disclosed herein. The edge support structure preferably is disposed on an internal surface of the crown so that the conformal weight is invisible when the golf club head is viewed at address. The conformal weight can be removed to adjust the vertical center of gravity of the club head, and preferably is oriented in a front-to-back direction along the club head's X-axis. The conformal weight preferably is composed of a high-density polymeric material with a specific gravity ranging from 1.8 to 4.2.