"Bear" [note spelling] doesn't sound right in that sentence. "Tolerate" is the basic meaning of "put up with". "Bear" is most often used with "can't". For example, "I can't bear harassment or bullying." rather than "We don't bear harassment and bullying at our school." That is not natural. "We don't put up with/tolerate harassment or bullying" is much better.
One difference is that "to bear something" it must happen to you. To "put up with or tolerate something", it could be happening to someone else.
Also used with 'can't', but generally not with 'don't', is 'stand': "I can't stand bullying." Yes. "I don't stand bullying" No.
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