Getting Potato and Turnip Patties to hold together... how?
My guesses:

First, if you're burning the outside before the middle cooks, you need to reduce your heat some. The outside will still brown over time at a lower heat, and the inside will cook better. If you start in and it looks like they're browning real fast, just turn the heat down (and maybe take the pan off the heat) until things get back under control. Then resume cooking at the lower temp. Getting the middle more thoroughly cooked should really help with them staying together, as the egg needs to be cooked to be an effective binder.

Second, make sure you drain the veg thoroughly before mashing. Any excess liquid may make them tend to fall apart. When they're drier they're going to be a little more stick-together-y in the first place, and then the cooking with the egg should bind it.

If neither of these things helps, I'd probably go for more egg rather than more flour. But I bet the other two options will sort you.