Dental insurance is usually pretty decent in the US, probably not worth leaving the country unless it's significant work. I have no idea with OP is getting done that costs thousands of dollars. Certainly that's an outlier.
At least for anything students can work on (fillings, extractions, etc). If you need a root canal, ortho, jaw surgery, or etc then I doubt it'll be a great option.
My dentist (in UK) charges £2500 for an implant. I don't have insurance.
(technically I do have insurance, since NHS dentistry exists which I pay for in my taxes. However I had an NHS dentist and when they got sold on to Bupa they would simply not do things which would be expensive for them, meaning after some years I had fillings falling out and broken teeth all over)
Dental insurance in the us usually only covers a max of $1000