air one's dirty linen in public
To talk about your private quarrels or disgraces where others can hear; make public something embarrassing that should be kept secret.
Everyone in the school knew that the superintendent and the principal were angry with each other because they aired their dirty linen in public.
No one knew that the boys' mother was a drug addict, because the family did not wash its dirty linen in public.
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Source: A Dictionary of American Idioms