"I am tired."
"is" is used when comparing two things to each other, see (tips & notes):
It is used to say that two things are equal, or to describe that one thing belongs to a class or category of other things: for example, in sentences like I am a man, the woman is a cook, he is our friend, or that is a book.
Also the answer here in the comments section isn't the answer they give when I got it wrong. That was "Tá tuire orm" here they say "tá mé tuirsach" is correct, I answered "Tá tuirsach mé" which they say is wrong, so which is it?
I am in deep water now, so anybody knowing better feel free to correct me:
The pronouns (mé, do, sí ...) and "tá" (or other verbs) are not really ever separated, the same applies to other forms such as "bhfuil" or "níl". "Nílimse tuirseach" or "An bhfuil tú tuirseach?".
This is related to the word order: Verb, subject etc.
I unfortunately am not skilled enough to find an answer to the word order definitely prohibiting "mé" at the end of any "tá" sentence without anything been allowed inbetween, but the basics is outlined in the two links below. (I would just kindly ask you to believe and to comply, even though that is not didactically correct. Especially the numerous samples of the second link don't support the suggestion of "Tá tuirseach mé".)
"Táim tuirseach." , "Tá mé tuirseach." and "Tá tuirse orm." would seem to be perfectly acceptable and having the same meaning. That again is an issue of languages, that you have a certain degree of freedom to say issues in different ways, no problem there.
(One place where the word order issue might be answered satisfactorily is this link, but that was way over my head:
I said "ta tuirseach orm" which we always said in school, But they gave the correct answer as " Ta tuirse orm" which is even different from the above???
tuirseach is an adjective - táim tuirseach
tuirse is a noun - tá tuirse orm.
In Ulster Irish, the pronunciation of tuirseach and tuirse isn't all that different.
It might help to translate the sentences literally: 'Tá mé tuirseach' - I am tired. 'Tá tuirse orm' - There is tiredness on me. Tired / tiredness.
In other parts of Ireland such as the southwest coast, we say "traochta" instead of "turseach"