In Usa there are legal ways to avoid it
Not in Poland there aren't, so why bother mentioning the USA? Even if he leaves Poland and lives elsewhere he is still liable for the payment of child support under Polish law.
@Saffalack, if an order for 'alimenty' has been made by the courts, you need to pay it. If you disagree with the amount, you can bring a case and appeal it but keep paying in the meantime or you will not be well received by any judge if you DO lodge an appeal - and you will have bailiffs (komornik) on your back at some point. If there is no order for support and your wife is simply expecting you to pay something, she has every right to expect that of you, both morally and under the law. However, if she is being greedy about the amount, then simply give her a figure you feel is affordable for you and fair to the child. Make sure that you have some proof of payment. Give her the money via bank transfer each time and use the reference 'child support' so that you can produce it in court as proof that you are meeting your responsibilities.