M.P.
I'm familiar with this sort of fabric. The sheet you put in the washer may catch the color while the fabric is in the washer but unless all of the dye is washed out the color will still bleed when damp/wet.
In the olden days we soaked fabric in a vinegar solution or perhaps it was a salt solution to set the dye so it wouldn't fade but I don't know if it would work on fabric that bleeds when wet. The fabric we soaked was colorfast and did not bleed but would gradually fade with washing. You could try it with a small swatch.
Doing this would definitely not keep the colors from bleeding into each other. I suggest that there really isn't a way to make these color fast.
You could, perhaps, sew them onto a plain piece of fabric so that the color isn't directly on whatever you put the fabric down on, say for a bed covering and then use dry cleaning to clean them. I could see using them for curtains or a valance or for decorating the edge of a canopy cover.
Here is a site that talks about salt and vinegar for setting dye and it says this doesn't work. Suggests trying a dye fixative but also suggests this isn't a known solution. No other suggestions. http://laundry.about.com/od/laundryproblems/f/How-Can-I-S...
Here is where to buy a dye fixative from Amazon. It might be worth a try if you truly love the fabric. http://laundry.about.com/od/laundryproblems/f/How-Can-I-S... Amazon lists 3 links for stopping dye bleed.