In some 40 years of quilting I've only had one bad experience with fabric bleeding. I've always been a believer in prewashing and even more so after having a top ruined. I actually did prewash the bad (name brand quilt shop quality) fabric but it wasn't sufficient It never did stop bleeding and turned a sophisticated blue, maroon,silver and white quilt into blue,maroon, silver and baby pink.
Prewash with color catchers and double check especially reds and dark batiks. You can take a little snippet of fabric (after the initial prewash), wet it and place on a white paper towel to see if any more dye is cast off.
I love a little bit of original Dawn. I've down other things including heat treating (boil or microwave), vinegar, salt, etc. but really, if it is still bleeding after 2 washes, I'd just move on to a more trustworthy fabric.