A club shouldn't have to publicly back their manager. Aside from it being a hollow endorsement, if a manager is underperforming, it's up to him to win their confidence by improving the situation, not the board/owner's to instill the confidence to turn things around.
Unless it's a situation in which it's a case of an otherwise trustworthy manager going through a bad period in the job, it's wrong of the club to back the manager and then be shown up later on for being perceived as hypocrites when they feel compelled to make a change.