You're going to have to talk more with your child and start some real nutrition ed. Not just "we don't eat this" or "this isn't healthy," but explain what these things do in the body, and why it's not healthy, and why you don't want those things in her body.
My daughter is 3.5, and she asks now if there is corn syrup in food she's given; she's aware that McDonalds is bad food that can hurt people, and that being too fat can hurt your body in ways x, y, z. (And she spreads the gospel, boy howdy.

) She's been playing with my molecular modeling sets since babyhood, so we talk about the molecules in food and how they go through your whole body, and how they get there. She's got a human-bio kids' book, too, with lots of pictures. We talk about how the body makes different kind of molecules into muscles and bones and teeth; how if you want to have a healthy body you have to put good stuff into it. How the body makes other food into sugar that goes in the blood, and how it goes all over the body to give us energy, but how too much sugar in the blood can hurt you. Basically she gets some very baby nutrition and human bio classes.

We also talk about how food is grown; we live across the street from a farm.
Anyway, she wants to be healthy and take care of the earth, partly because she sees these things are important to me. Knowing how things work lets her make her own decisions when food is offered to her. So the other night, for instance, at a restaurant, the waitress offered her fries, chips, or fruit, and she picked fruit without any prompting from me. And she talks often about having a healthy body. As she gets older I'm sure she'll be a less ready disciple

, but she'll have the info and be able to use it & ask questions.