As I read this my first thought is that your family loves to eat food prepared by a quick in-and-out restaurant or fast food place.
Honestly, you can make all of those things at home from scratch and by themselves none are necessarily "unhealthy."
Hamburgers. Start with lean ground beef (93/7 sirloin and it has a great flavor) and you've already cut the fat and grease of a fast food place probably by half. Then, grill it on a grill where much of the oil and fat drip out. Choose a whole wheat or whole oat bun over white/sugar buns, or go without the bun completely. Top it with fresh veggies - tomato, lettuce, avocado, sautéed onions... all healthier options than McDonald's offers.
Fries: anything baked in the oven is immediately healthier than fried in vegetable oil. My family loves traditional fries in the oven. Cut a potato (I leave the skin on, actually as it is easier), toss it lightly in olive oil, sprinkle with seasoning (salt, pepper, sometimes chili powder, you can be creative here) and bake it at 400 for about 15 minutes, then flip them and give them another 10 to 15 minutes (depending on the size of your cuts). You can do the same thing with sweet potato for an even healthier option, that is also delicious.
Pizza: There are so many options for healthier pizza when you make it yourself. It starts with the base (or the "dough"). You can always experiment with different more healthy bread options. But last summer at the peak of zucchini season, I actually made pizza bites on zucchini and it was divine. Of course cutting the pepperoni and other fatty meats from the top raises the health factor - consider Greek pizza: Mediterranean olives, sun dried tomatoes, artichokes, feta cheese, spinach.
I think all of these things can keep your family eating what they like, but they sort of raise the bar on health and flavor, you know? Turn your family into foodies!