Green’s Dictionary of Slang

screw up v.

1. [late 17C–19C] to cheat, to defraud; to drive a very hard bargain.

2. [early 19C] to imprison [screw n.1 (2)].

3. [mid-19C; 1940s+] to make trouble, to cause difficulties for.

4. [mid–late 19C] to garrotte [SE screw up, to tighten].

5. [1920s+] (orig. US) to make a mess, to blunder badly.

6. [1940s] (US) to confuse, to perturb.

7. [1950s+] to hurt, to put out of order, to destroy.

In phrases

screw up (with) (v.)

[1960s] (US) to get involved with, associated with.