Thurston was born in Montpelier, Vermont. He moved with his parents to Madison, Wisconsin, in 1854 and two years later to Beaver Dam, Wisconsin. He attended the public schools and graduated from Wayland University in Beaver Dam, where he studied law. Thurston was admitted to the bar in 1869 and commenced practice in Omaha, Nebraska. He was a city councilman 1872-1874 and the city attorney of Omaha 1874-1877. Thurston then served in the Nebraska House of Representatives 1875-1877.

He was appointed assistant attorney of the Union Pacific Railroad in 1877 and general solicitor in 1888. He was a presidential elector on the Republican ticket in 1880. In 1893, Thurston was an unsuccessful Republican candidate for United States Senator; he was elected as a Republican to the United States Senate two years later and served from March 4, 1895, to March 3, 1901. During his term, Thurston served as the chairman of the U.S. Senate Committee on Indian Affairs (Fifty-sixth Congress). He was not a candidate for reelection in 1901. He was appointed as the United States commissioner to the St. Louis Exposition in 1901. He moved to Washington, D.C., and resumed the practice of law; some time later, Thurston returned to Omaha and practiced law until his death on August 9, 1916. His remains were cremated at Forest Lawn Cemetery in Omaha and the ashes were interred in the Congressional Cemetery, Washington, D.C.

Thurston County, Nebraska is named after him.