"If I study internationally, would I have trouble practicing in the U.S. as an orthodontist? "
the problem won't be practicing orthodontics in the US, but rather it will be GETTING to be an orthodontics in the US.
US ortho schools are famous for the difficulty to get in. of course they prefer US graduates, but if you're a foreign graduated with a great record you have a good chance. I mean, great GPA, honors, great boards scores, 1 year residency program, etc.
US ortho schools prefer high numbers in anyone, no matter where they come from. So, now that you know, no matter where you study, you better want to start to prepare early for such a competitive process.
but, musicmaker, have you applied to US or Irish schools? Do you have the possibility of going to a US school? Things are way easier if you study here.
good luck!