"저는 동료가 많다고 말했습니다."
Translation:I said I have many colleagues.
In English, tense backshifting in reported speech/indirect quotes is applied most of the times, but not always.
• optional if the situation is still true
• not applicable for general truth.
Avoid backshifting if the original verb (tense used in the direct quote) is
• in past perfect
Backshifting is optional in the given example. Neither you nor DLG is wrong. Just a matter of DLG accepting your suggestion as an alternative answer.
Korean has a very different syntactic structure from English so having only one "correct" translation is pretty blasphemous - the 고 morpheme translates as "that" directly and "many" and "a lot" mean the same thing, so my translation of "I said that I have a lot of colleagues" should be correct (and is a more natural English translation in my opinion).