Is there a misspelling or I am mistaken and "agzim" is correct instead of "agizim" (no TR keyboard available, sorry)
As you've probably seen in some words before there is a vowel omission there. So 'ağzım' is correct.
Well we've seen this happen with the word for city, so it's not without precedent. A native speaker will have to confirm though.
But why should we omit a vowel inside the word? mean the 'i' after the 'g' is still included in the noun itself. I thought we only omit the last letter if it were a vowel.
oğul - oğlum (son - my son), şehir - şehri (city - city Accusativ), ağız - ağzım (mouth - my mouth) As you can see in all these cases it is a vowel inside the word.
also, to add to this, rules like vowel omission or consonant changes are always applied to ease pronunciation. for example, if we would say "ağızım" instead of "ağzım", not only we would have more syllables, we would have a lot more oral movements, which would make it harder to talk fast.