It sounds to me like the issue may be that you're crowding the pan.
Basically, to get everything nice and brown and crispy, you need enough space for all of the steam to escape. That picture you showed has potatoes stacked on top of each other -- that means as the bottom items cook, they're going to end up steaming the items above them.
At a diner, they have a large griddle to work with -- they can really spread things out. You're not typically that lucky in a regular kitchen, as you don't have as much space, and you have a lip on the pans that'll hold the steam in.
So, either work in smaller batches, or consider recipes that use an oven -- using sheet pans instead of a pan on the stove solves much of the problem.
One other trick is that most diners don't start from raw potatoes -- maybe with hash browns, but not for home fries, you're not going to get the nice soft interior in a reasonable amount of time unless you start with a potato that's already been baked or boiled. (If you're doing things in the oven, you might be able to, but not in a pan)
Just for reference ... I have a 14" cast iron skillet that I use for home fries ... and it's about the right size for cooking a single large potato, which might be two servings, maybe three for kids. (I tend to cook carb-heavy meals).
update : I probably should've stated this directly -- you want the chunks of potato to form a single layer in the pan, with space in between them.
You need to cover soft vegetables in egg or batter first. Normally the vege's are steamed first to actually cook, cooled, then egg washed or lightly battered and flash deep fried in hot oil just to set the thin batter
Try Indian style spicy pea/chickpea flour batters
Best Answer
I assume you are referring to standard, grocery store, cavendish bananas-
They have a ton of water and sugar and have a very delicate texture. This makes them easily disintegrate and get gummy instead of crisp. The key is to deep fry them at a fairly high heat until they are dark and crisp. It also helps to use greener bananas that will hold together better.
Butter is not well suited to this because of the necessary oil depth and high heat. Not only would it take a fair bit of butter but the butter would start to burn. Even clarified butter wouldn't be ideal.
Vegetable oil or shortening would be the way to go.
Additional notes:
To make frying them easier bananas are often coated in something starchy- like a tempura batter. Recipes abound and they are, in my opinion, much better than frying the bananas on their own.
Plantains are more often fried because they are much more starchy than Cavendish bananas. Plaintains actually become sweeter when fried.