Norway is a town in Orangeburg County, South Carolina, United States. The population was 337 at the 2010 census. Norway was laid out in 1891 when the railroad was extended to that point. A post office has been in operation at Norway since 1892. The town was so named in order to fit with the railroad's "Scandinavian" naming scheme; other such examp…Norway is a town in Orangeburg County, South Carolina, United States. The population was 337 at the 2010 census. Norway was laid out in 1891 when the railroad was extended to that point. A post office has been in operation at Norway since 1892. The town was so named in order to fit with the railroad's "Scandinavian" naming scheme; other such examples include Sweden, South Carolina and Finland, South Carolina.