At least in US criminal court procedure, each side generally has a limited number of "peremptory challenges". These can be used to excuse (dismiss) potential jurors without specifying any reason, and it is very unusual for either side to specify the reason. Such challenges may not be used to exclude jurors solely on the basis of race, as the answer by Jen correctly explains in detail. It has been suggested that this limitation also be extended to other protected characteristics such as sex or ethnicity, but I am not aware of a case that has imposed such a restriction. Even the prohibition of racial exclusion can be hard to enforce, if the attorney, when questioned, asserts some other basis for a particular challenge.
Aside from the limited number of peremptory challenges, either side may "challenge for cause". This means alleging some reason why a juror would not be qualified, or would not be properly impartial. A source of possible bias would be a classic reason for a challenge for cause. But all such challenges are reviewed by the trial judge (or judge supervising jury selection, in the unusual case where this is not the trial judge), and a juror is only excluded if the judge agrees that the challenge is valid. So an attorney can suggest that people who support, or strongly oppose, Nazi parties are biased, but the Judge may well not accept this and may choose to retain such jurors. It is a judgement call.