NGINX must be setup properly: it does not work out of the box. Check out my tips on modular NGINX configuration. For your specific issue, you must adjust the root directive.
If for some reason you do not have access to NGINX configuration, this is going to become really ugly. Your best bet is to disable the public folder entirely (remove public and do a composer update), but I think you are going to face other problems. I strongly suggest to gain access to the configuration files in some way.