"Why do you start the computer?"
Translation:Warum startest du den Computer?
Does "Startest du den Computeromputer" mean you turn on the computer?
No. Your sentence starts with the verb, so it's a yes-no question rather than a statement: "Do you turn on the computer? Are you turning on the computer?"
A statement would have the verb in the second position: Du startest den Computer.
"to turn up" means to make louder/higher, or suddenly appear; not something generally used in terms of computers.
"to turn on" should work, though. But that is better represented by "einschalten" in German.
starten = to start, to start up, to power up, to boot up (computer specific)