You work in NJ but NY state tax is withheld. That seems odd. NJ income normally means NJ tax is paid. Then NY gives a credit for the double taxed income - the lower of the state taxes on the same income. NY tax normally due a little.
If NY tax is being withheld from each check and you still owe, then failure to withhold enough tax from your paycheck is the issue. You need to file a new w4 with your HR department to have more withheld
In addition, you may have city tax added on to your state return.
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