You are pointing to public as document root? Main laravel files should be out of (one folder above) public_html.
FYI, shared hosting has be covered many times. Try punching this into Google:
site:laracasts.com set up laravel on shared hosting
Also I believe that host has how to articles.
Also another guide: http://novate.co.uk/deploy-laravel-5-on-shared-hosting-from-heart-internet/
One past post: https://laracasts.com/discuss/channels/laravel/how-to-deploy-laravel-9-on-shared-hosting
But many more.