I had turkeys with chickens for years with no problems; blackhead was not in the soil. I've retired to a smaller place in another state and blackhead is present here. We have a specialist avian vet here, so I got his advice and information. Getting rid of the chickens won't make a difference. The blackhead organism is a protozoa parasite of cecal worms, which are a parasite in earthworms. When it's muddy and the turkeys are eating earthworms that come to the surface to avoid drowning, they will get the parasites if blackhead is present in your soil.
Young birds are most susceptible; as with coccidia (which are present everywhere) they develop some immunity/resistance as they mature, if you can prevent them from being overwhelmed by large numbers of the parasites when they are first exposed. Young poults first put out on pasture will become infected, look sick one day and be dead the next. Medicated chicken feed containing amprolium is a coccidiostat (stops coccidia from reproducing, doesn't kill them), and has no effect on blackhead. Also, if you have waterfowl, like I do, amprolium will kill them.
If you can eliminate or reduce the number of cecal worms in your turkeys, you are reducing the numbers of blackhead parasites they can transmit. Cecal worms live in the cecum (surprise surprise), but the blackhead parasites migrate into the liver. The characteristic sign of blackhead, other than sudden illness and quick death in poults, is yellow droppings from compromised liver function in adults, and the birds act like they are not feeling well.
You can control/reduce cecal worms in your turkeys when you have muddy conditions, with Ivomec by mouth. 1/10 to 3/10s of a cc of the cattle liquid, depending on size of turkey. Ivomec has a large safety margin but don't go crazy. The vet said Panacur could also be used but I don't know the dose, but do NOT use Valbazen as it will kill the turkeys. I repeat the Ivomec dose every 10 days during my wet seasons, and this has worked well to prevent or reduce the blackhead infestation. I have not lost any poults since I started doing this. Adult birds with blackhead are less likely to die suddenly, but look ill and develop yellow droppings. Any turkey that starts having yellow droppings should be treated for blackhead directly. As a protozoan it is sensitive to the Ronidazole, Dimetridazole type of medications. You could buy powder to put in the water, but you would have to isolate the bird and keep track of how much it consumes. I had a sick adult tom (when I first moved here, that's how I learned all this from the vet).
Because I also have pigeons, I knew that both drugs were available in pills intended to treat pigeon canker. These are readily bought online at Pigeon Supplies Plus, or other suppliers. Either drug works, you have to calculate the dose based on the dosage of the pill and your turkey's weight in kilograms. After doublechecking with the vet on the dose, I used Ronidazole 30mg. pills and gave my Royal Palm tom 3 pills by mouth every evening. No, he didn't like it. I am a small person, I found it easiest to do this by kneeling over him to push him in a sitting position on the floor, pry open his beak and drop the pills way down his throat (behind the back of the tongue). Because this was treatment, not prevention like with the Ivomec, the vet recommended 14 days of treatment. He did recover, and the vet tested him to see if he was free of parasites afterward, and he was. However, he sustained some liver damage and he will always have yellow droppings. This was 2 years ago, and he's fine. By worming the other turkeys during muddy seasons, none of the others have been infected with the blackhead parasite. My birds are handled often, so they are pretty easy to catch and medicate as needed, which is only a few times a year, less often than worming a horse or cow (or outside dog). I am able to worm all of my turkeys by holding them with my left arm wrapped around the body, their feet resting on my bent leg, and stuffing a small plastic syringe (no needle!) with Ivomec into their beaks. Again, I am a very small person and not strong.
You could decide to stop keeping turkeys if you have blackhead in your area, but I've had mine for 15 years and enjoy them tremendously and I'm not giving them up. If you have it in your soil, you might be able to keep adult turkeys who have developed a resistance, but you are almost sure to lose your young ones as they go out on pasture.
I hope this helps.