wash (someone's) dirty linen in public

Translations
выносить грязное белье на публику
They decided not to wash their dirty linen in public and resolved their issues privately.
Они решили не выносить грязное белье на публику и разрешили свои проблемы в частном порядке.
Definitions
To discuss or reveal private, personal, or embarrassing matters or disputes in front of others, especially in a public setting.