How do I make nginx stop showing "Welcome to nginx!" and show a site as default?

First, you can remove or replace that default nginx welcome page. It’s a static index.html file located in /var/www/html.

The following code snippet will redirect any requests for your IP to the domain (after domain.tld is replaced with your domain and [ip address] is replaced with your droplet’s public IP]:

server {
    listen 80;
    server_name [ip address];

    return 301 $scheme://domain.tld$request_uri;
}