Bursum, was born at Fort Dodge, Iowa to Norwegian-American parents. He attended the public schools in Iowa before moving to New Mexico Territory in 1881. He settled near Socorro and engaged in raising livestock. He was a member of the New Mexico Territorial senate, 1899-1900; chairman of the Territorial central committee in 1905 and 1911; member of the State constitutional convention in 1910; and a member of the Republican National Committee, 1920-1924.
Bursum was appointed on March 11, 1921, and subsequently elected on September 20, 1921, as a Republican to the United States Senate to fill the vacancy caused by the resignation of Albert B. Fall and served from March 11, 1921, to March 3, 1925. He was an unsuccessful candidate for reelection in 1924. He served as chairman, Committee on Pensions (Sixty-seventh and Sixty-eighth Congresses) engaged in the newspaper business at Washington, D.C., and subsequently returned to Socorro and resumed his former business interests until his death in Colorado Springs, Colorado. He is interred in Socorro Protestant Cemetery in Socorro.