DeShannon's biggest break came in February 1964 when she supported The Beatles on their first US tour, and formed a touring band with guitarist Ry Cooder. DeShannon co-wrote "Breakaway" with Sharon Sheeley, which was recorded by Irma Thomas in 1964, and by Tracey Ullman in 1983. She also wrote "Don't Doubt Yourself Babe" for "Mr. Tambourine Man", the 1965 debut album of The Byrds. Her music at this stage was heavily influenced by the American West Coast sounds and folk music. Staying briefly in England in 1965, DeShannon formed a songwriting partnership with Jimmy Page, which resulted in the songs "Dream Boy" and "Don't Turn Your Back on Me". DeShannon also wrote material for singer Marianne Faithfull, including her Top Ten UK and US hit "Come and Stay With Me", which became Faithfull's biggest UK hit, peaking at No. 4 in 1965. That same year, Cher covered the song on her solo debut album "All I Really Want to Do". It would be three more years before Jackie DeShannon would record the song for herself, on her 'Laurel Canyon' album in 1968. She also appeared on the television show "Ready Steady Go!"