It's tough to find cat food without byproducts in the grocery store brands, but at least in the US, there are many foods available with no byproducts. I would also steer clear of corn, wheat and soy since those are common allergens for cats.
You can find some cat foods without byproducts, grains or soy at Petco or Petsmart, or order online from sites like petflow.com or petfooddirect.com You can search and check ingredients on here
http://www.petflow.com
I've got a cat who has IBD and get very bad reaction ( vomiting blood and diarrhea, & has had to be hospitalized ) from some foods like corn and beef.
The vet ( internal med specialist) recommended a grain free food with venison, rabbit or duck (a type of protein he was not used to. The vet said don't just switch constantly between all those kinds of meat in case he develops a reaction to one, so he could then try another ) . Nature's Variety Instinct makes foods in those kinds. I chose the venison , and feed mostly canned and some raw frozen. They also have dry food including a rabbit version. ( but cats do need to eat at least half wet food since they dont tend to drink enough to keep well hydrated if on mostly dry)
Unfortunately the NV Instinct is rather expensive. The cat and dog canned formulas are the same thing, just different labels. and it's more economical to get the 13 oz can, but still pretty expensive.
You may end up needing to feed one of those (or lamb.) or make your own cat food. There is a recipe on Dr Lisa Pierson's site
http://catinfo.org , which is a good place to start learning more about cat nutrition.
But right now if what you're trying at first is just to avoid byproducts, there are lots more possibilities.
( though I really would suggest avoiding the corn, wheat and soy. and don't feed fish very often. )
Some other brands Merrick, Merrick Before Grain, Wellness, Blue Buffalo Wilderness , Weruva, Innova Evo,
.