Kurtz made his conducting debut when he substituted for an ill Nikisch to accompany the dancer Isadora Duncan on tour. This led to a number of concerts with the Berlin Philharmonic. From 1924 to 1933 he conducted the Stuttgart Philharmonic, and in 1928, Kurtz was enagaged by Anna Pavlova to accompany her dancing, which he did until her death in 1931. From 1932 to 1942 he was conductor of the Ballet Russe de Monte Carlo, touring with them extensively. His work in Monte Carlo included conducting the premiere of Gaîté Parisienne.
Kurtz later moved to the United States, and became a citizen of that country in 1944. He was music director of the Kansas City Philharmonic from 1943 to 1948. He held the same post with the Houston Symphony from 1948 to 1954. He also conducted a number of film scores, including Jacques Ibert's score for Orson Welles' Macbeth.
From 1955 to 1957, Kurtz was music director of the Royal Liverpool Philharmonic jointly with John Pritchard. Thereafter he took a number of guest conducting posts, including engagements with orchestras in Leningrad and Moscow in the Soviet Union where he returned for the first time in 1966.
Kurtz was married three times. He was married to the American flautist Elaine Shaffer until her death in 1973. After her death, Kurtz married Mary Lynch, who survived him.