Turtles and fish generally make poor roommates. Occasionally you can get a turtle and fish of similar sizes to share a tank without incident. But predation will usually cause problems for your fish, as most turtles will eat most aquarium fish. Concerns over disease are a distant second to concerns over turtles eating fish when these critters share an aquarium.
Turtles and fish are very different animals. For the most part, diseases that infect one don't harm the other. However, a few exceptions exist. Some opportunistic diseases like fungi can infect both fish and turtles. Opportunistic infections occur when a fish or turtle gets hurt, injured or sick. In this state, normally harmless organism, present in most aquarium water, can suddenly cause infection. Both turtles and fish may develop such infections. However, these infections happen most often in dirty, poorly maintained aquariums. If you keep up on water changes, theses diseases rarely occur.
When keeping fish and turtles together, turtles' predatory nature presents more cause for concern than disease. Almost all turtles are at least somewhat predatory, particularly aquatic and semi-aquatic species that can live in an aquarium. Most turtles will eat fish smaller than themselves, and some more pugnacious species may take bites out of larger fishes.
Very small turtles may not prey on fish, but most pet turtles will continue to grow. And sooner or later, fish will start to go missing. In many regions, it's illegal to purchase turtles under a certain size due to salmonella concerns. Still, if you have a larger tank you can move the turtle to as it grows, you could keep a young turtle with fish for a time. If you have a very large aquarium in the first place, you can keep larger fish, about the same size as the turtle, in the same aquarium. Research your exact species of turtle and fish to make sure neither has a tendency to take bites out of larger organisms.
Turtles also have a few care requirements above and beyond those of aquarium fish. Most turtles need a place where they can crawl out of the water, as well as a heated spot for basking, to help the turtles regulate their body temperatures. Turtles need special reptile lights. These lights produce a little light in the UV spectrum, which turtles need to make vitamin D. Since turtles have these concerns, it's easier to keep turtles in a separate tank from fish and enjoy these two pets separately.