Thestrup had a history of alcohol abuse, which contributed to his 1988 imprisonment at Headingley Correctional Institution near Winnipeg, Manitoba, Canada after he caused disruption on board a British Airways flight from London to Los Angeles, leading to an emergency landing.

[3] Thestrup's early television roles included the comedy series Een stor familie (1982–1983) and the 1986 Christmas calendar show Jul på Slottet, in which Politiken described him as being "perfectly cast".

[5] Director Anders Thomas Jensen regularly cast Thestrup in his films, beginning with the Academy Award-winning 1998 short Election Night.

He also appeared in Jensen's feature films Flickering Lights (2000), The Green Butchers (2003), Adam's Apples (2005) and Men & Chicken (2015).

Thestrup had a recurring role in all three series of the DR political drama Borgen as right-wing politician Svend Åge Saltum.