Southwest Bridge Research Center Establishment Act of 2003 - Amends Federal transportation law to direct the Secretary of Transportation, in addition to grants that university transportation centers receive under the Act, to provide grants to New Mexico State University, in collaboration with the Oklahoma Transportation Center, to establish a university transportation center known as the Southwest Bridge Research Center. Declares that the purpose of the Center is to contribute at a national level to a systems approach to improving the overall performance of bridges. Sets forth certain grant requirements.