Nginx proxy: redirect to http://private-service/

I'm trying to configure a nginx reverse proxy. I'm using htpasswd but it's unrelated in my case.

I want the following: when visit https://public-nginx/kibana I want that it redirects to the proxied url (to my private kibana), but it keeps the public-nginx URL.

My nginx.conf looks pretty basic:

worker_processes 1;
error_log /var/log/nginx/error.log;
pid /var/run/nginx.pid;
include /usr/share/nginx/modules/*.conf;
events {
    worker_connections 1024;
}
http {
    log_format  main  '$remote_addr - $remote_user [$time_local] "$request" '
                      '$status $body_bytes_sent "$http_referer" '
                      '"$http_user_agent" "$http_x_forwarded_for"';
    access_log  /var/log/nginx/access.log  main;
    server_names_hash_bucket_size 128;
    sendfile            on;
    tcp_nopush          on;
    tcp_nodelay         on;
    keepalive_timeout   65;
    types_hash_max_size 2048;
    include             /etc/nginx/mime.types;
    default_type        application/octet-stream;
    include /etc/nginx/conf.d/*.conf;
    index   index.html index.htm;
    server {
        listen       8080 default_server;
        listen       [::]:8080 default_server;
        server_name  localhost;
        root         /usr/share/nginx/html;
        include /etc/nginx/default.d/*.conf;
        location / {
        }
        error_page 404 /404.html;
            location = /40x.html {
        }
        error_page 500 502 503 504 /50x.html;
            location = /50x.html {
        }
    }
}

my conf.d/kibana.conf looks like:

server {
    listen 80;
    server_name _;
    location / {
        proxy_pass https://private-url/kibana/;
        proxy_redirect https://private-url/kibana/ /;
        proxy_http_version 1.1;
        proxy_set_header Host $host;
        proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
        proxy_set_header X-Real-IP $remote_addr;
        proxy_set_header Authorization "";
        proxy_hide_header Authorization;
        auth_basic "Username and Password are required";
        auth_basic_user_file /etc/nginx/.htpasswd;
    }
}

When I check it seems that the /kibana url is redirecting me to the right path (/kibana/app/..). But I see a 404.