If you go to the Avatars tab in your Profile, select a desired picture, and press the Select button, but your avatar doesn’t change, you most probably have a special avatar from the Scatter Pass Perks set up. To fix the issue and change your avatar to one of the common or rare ones, you need to disable the one from Scatter Pass.

To do that, please follow these steps:

1. Go to your profile (tap your avatar in the top-left of the screen)
2. Tap the Scatter Pass icon (a yellow sword).
3. There, you can find the Scatter Pass Perks tab. To disable the avatar, you need to uncheck the green checkmark under the selected picture.

After you disable the fancy avatar in this menu, you will be able to select any other avatar you like (from common or rare ones in your Profile).