@littleadv has good advice here.
You most likely owe taxes to both countries, but the question is "how much?"
I have this to add:
IRS Publication 54 is about the US taxes of Americans living overseas.
In particular, Chapter 2 covers the topic of employer withholding, and mentions a Form 673 you might be able to file with an employer to reduce withholding if you are in fact eligible for the large Foreign Earned Income Exclusion that exists in the understanding that living in a foreign country means you pay foreign taxes already. Basically you need to be able to prove that you really live in a foreign country and pay taxes there, as opposed to merely owning property or being a visitor there. An alternative way to qualify is to spend 330 days per year in a foreign country.
Here is a relevant excerpt of the IRS publication:
Income Tax Withholding
U.S. employers generally must withhold U.S. income tax from the pay of
U.S. citizens working abroad unless the employer is required by
foreign law to withhold foreign income tax.
Foreign earned income exclusion. Your employer does not have to
withhold U.S. income taxes from wages you earn abroad if it is
reasonable to believe that you will exclude them from income under the
foreign earned income exclusion or the foreign housing exclusion.
Your employer should withhold taxes from any wages you earn for
working in the United States. Statement. You can give a statement to
your employer indicating that you expect to qualify for the foreign
earned income exclusion under either the bona fide residence test or
the physical presence test and indicating your estimated housing cost
exclusion. Form 673 is an acceptable statement. You can use Form 673
only if you are a U.S. citizen. You do not have to use the form. You
can prepare your own statement. See a copy of Form 673, later.
Generally, your employer can stop the withholding once you submit the
statement that includes a declaration that the statement is made under
penalties of perjury. However, if your employer has reason to believe
that you will not qualify for either the foreign earned income or the
foreign housing exclusion, your employer must continue to withhold.
In determining whether your foreign earned income is more than the
limit on either the foreign earned income exclusion or the foreign
housing exclusion, if your employer has any information about pay you
received from any other source outside the United States, your
employer must take that information into account.