He represented Ireland in the Triangular Professional Tournament in 1937 and Llandudno International Golf Trophy in 1938.

He won the Irish PGA Championship 10 times between 1941 and 1957, tied with Christy O'Connor Snr for most wins in that event.

He teamed with Christy O'Connor to win the Canada Cup for Ireland in Mexico in 1958, finishing second in the individual section of the event despite suffering nosebleeds due to the altitude.

Bradshaw lost the 1949 The Open Championship following a playoff against Bobby Locke at Royal St George's, after an extraordinary incident in the second round when his drive at the 5th hole came to rest against broken glass from a beer bottle on the fairway.

[1] PGA Tour playoff record (0–1) Note: Bradshaw only played in The Open Championship.