Important question: are you bleeding from the gums when you floss?
Healthy gums can handle normal flossing without bleeding. I floss once a day, before bed. Normal flossing does not involve super hard scrubbing, just enough force to scrape off stuff stuck between the teeth and dislodge stuck particles. You might also want to add some antiseptic mouthwash to your routine after flossing until your gums stop bleeding.