There are exclusions for foreign income tax paid, but I believe there is a cap (US$75,000?) on the amount of tax paid. That is, tax paid to another country can reduce your US liability.
Anyone in this situation needs competent advice. That leaves me out as a source of that advice.
I first moved to the US in May of 1989. At the end of the year, when preparing my US tax returns, I had to report my UK income and UK taxes. The tax treaty was essentially reciprocal, so the final position was that the the US government treated the taxes I had paid the UK as part of my US taxes. Since the UK taxes at a higher rate, I actually got a refund from the US government!
I figured all this out by going to the library and actually reading the full text of the tax treaty between the US and the UK. It seems that most people wouldn't do this, and would pay "an expert" instead.
Along with Eritrea and North Korea.