I'm not sure what's going on here. In your view you are calling
<img src="{{Auth::user()->images}}"/> // images?
And then in your controller you are
User::where('id', Auth::user()->id)
->update(
[
'image' => $request['image']] // image?
);
Question is, what are you storing and how do you know if the image is stored correctly? If you are attempting to store the file path, you are not really storing the file anywhere. If you are attempting to store it as a binary to the database, I'm pretty sure that's also not the correct way (especially the output would be different).
Please take a look under the documentation to get a better understanding of uploading files using laravel: https://laravel.com/docs/5.8/filesystem#file-uploads