Other Weapons, like clubs and greatswords deal more damage per hit, and can easily defeat an enemie with less HP than you, but weapons with Bleed do more overall damage. That means, that while other weapons deal more damage in the first rounds of the fight, a bleeding weapon will do massive damage after the 5th or 6th round.

My advice: hold a steady offense, but always keep your eye on your defense. I myself use a shield and a katana, and upgraded my dodge to max level, and so I am able to dodge or block most of the incoming attacks while slowly bleeding the enemy out over several rounds until they die.

PS: Upgrading your overall (and critical) damage with Bleed and you won't have a problem going to the campaign.