When in any State of the United States under the irrigation district laws of said State there has, prior to August 11, 1916, been organized and created or shall thereafter be organized and created any irrigation district for the purpose of irrigating the lands situated within said irrigation district, and in which irrigation district so created or to be created there shall be included any of the public lands of the United States, such public lands so situated in said irrigation district, when subject to entry, and entered lands within said irrigation district, for which no final certificates have been issued, which may be designated by the Secretary of the Interior in the approval by him of the map and plat of an irrigation district as provided in section 623 of this title, are made and declared to be subject to all the provisions of the laws of the State in which such lands shall be situated relating to the organization, government, and regulation of irrigation districts for the reclamation and irrigation of arid lands for agricultural purposes, to the same extent and in the same manner in which the lands of a like character held under private ownership are or may be subject to said laws: Provided, That the United States and all persons legally holding unpatented lands under entry made under the public land laws of the United States are accorded all the rights, privileges, benefits, and exemptions given by said State laws to persons holding lands of a like character under private ownership except as in this chapter otherwise provided: Provided further, That this chapter shall not apply to any irrigation district comprising a majority acreage of unentered land.