Please follow the guide how to install Drupal 10

composer create-project drupal/recommended-project my_site_name_dir

You need to keep this structure, the "project-root" will be at my_site_name_dir and your webserver must point to the "web-root" which will be at my_site_name_dir/web by default, AFAIS you want to change only the directory of the web-root which you can easily do by adjusting the composer.json drupal-scaffold, e.g

"extra": {
        "drupal-scaffold": {
            "locations": {
                "web-root": "web/"
            }
        },

and make composer install

Please follow the guide how to install Drupal 10

composer create-project drupal/recommended-project my_site_name_dir

You need to keep this structure, the "project-root" will be at my_site_name_dir and your webserver must point to the "web-root" which will be at my_site_name_dir/web by default, AFAIS you want to change only the directory of the web-root which you can easily do by adjusting the composer.json drupal-scaffold, e.g

"extra": {
        "drupal-scaffold": {
            "locations": {
                "web-root": "web/"
            }
        },

and adjust the installer_paths accordingly

and make composer install