"I ran a lap" would be "Jeg løb en omgang."
Just like you say "I ran a lap" or "I ran 2 miles" in English, you can say "I ran a trip" in Danish => "Jeg løb en tur."
what about "I went to run"? If they anyway don't give a literal translation, then why "i went to run" is wrong???
It might be hair-splitting, but both "Jeg løb en tur" and "I went for a run" say that the runner made a round, a trip. Going for a run and coming back again.
Your version sounds like the runner is just running around aimlessly.
To me it sounds like I would only use that sentence in a specific circumstance.
Why were you in Philadelphia? Oh, I went to run (in a race).
I went to run, but he held me back.