My iPad mini keeps flashing back an forth to different screens

Your main question line and description in the text block are asking two different things. In one you are asking why the mini is flashing from screen to screen and in the other one you are asking why it is flashing from one site to another .... As in web site? So which is it? Different screens on the iPad itself? Different websites in Safari?

In either case I would clear Safari, close all apps and force restart your iPad.

Go to Settings>Safari>Clear History and Website Data

Now close all apps. Press the Home button two times quickly. You'll see small previews of your recently used apps. Swipe left to find the app you want to close. Swipe up on the app's preview to close it.

To force restart the device, press and hold both the Sleep/Wake and Home buttons for at least 10 seconds, until you see the Apple logo.

My iPad 4 does similar things, you can be on safari and all of a sudden it's like it has a mind of its own and starts opening new tabs and typing random things in,

it also acts strange in iBook and the Photo app, it'll change page on its own in iBook and even change book, very bizarre behaviour, very annoying too!!