Note the literal translation of the sentence. There is no direct equivalent of the verb "to have" in Irish, so rather than saying ""The turtle has carrots", we say "Carrots are at the turtle" (in other words, the carrots are close by and readily available to the hungry turtle). "Carrots" (cairéid) is the subject, and "are" (tá) is the verb.
Your version would translate as "The turtle is at carrots" - in other words, "Carrots have the turtle"! Unless we are talking about a sequel to the movie "Attack of the Killer Tomatoes", that's probably not what you meant...
Words that begin with 'd' or 't' are an exception to the rule that certain prepositions followed by the singular definite article ('an') trigger eclipsis.
Tá cairéid ag an gcat. (The cat has carrots.) Hence 'cat' becomes 'gcat' following 'ag an.'
Tá cairéid ag an turtar. (The turtle has carrots.) Hence 'turtar' does not become '*dturtar' following 'ag an.'
However - please, someone, correct me if I am mistaken - 'turtar' might eclipse in other situations; e.g., would 'nine turtles' not be 'naoi dturtar?' (and not 'naoi *dturtair' because 'na dturtar' is the genitive plural?)