I was writing a story for Wattpad, in which the two main characters spoke both Japanese and English, when I wanted one to say "You did no better than I". I looked it up in my dictionary, and I could not find the word 'did'. Another way to say that sentence would suffice, but I would appreciate how to say the word 'did' in Japanese.
The past tense of する (suru) would not show up in a dictionary as it is a conjugation of a verb. Conjugations typically do not show up in a dictionary. For that, you're better off consulting a textbook, to pick up verb conjugations as you learn the language.
That being said, the past tense conjugation, sometimes referred to as the た(ta)-form, for する (suru) is した (shita).
Also, for what it's worth, I'm not sure what tone your character is speaking in, but for formality's sake, there are separate conjugations for past tense depending on how polite you'd like to sound. した(shita) is standard, but for more polite, you might want to try しました(shimashita).