The URL of a website depends on the domain name and the extension to it (.com, .in, .org etc.). You can add a subdomain like a prefix (subdomain.yourwebsitename.com) or a sub-directory slug (yourwebsitename.com/slug) to the domain like a suffix. If you don’t want to do such thing and want to change the current domain name to a new domain name, then you can follow the steps:

i)      Buy a new domain of your choice.
ii)     Update the nameservers of your new domain in DNS management section.
    (use the nameservers of your hosting provider).

iii)    Take backups of the ‘Public_html’ and ‘MySQL Database’ folder of your old
    website/domain.

iv)    Upload the ‘Public_html’ and ‘MySQL Database’ folder backups of your old
    website/domain under the new domain.

v)    Connect the old database and public html folders to the new domain.

vi)    Edit .htaccess file in public_html and replace the old domain name with new domain name, so that the visitors to your old domain name can be automatically redirected to the new domain name. *I have give a sample htaccess file bellow.

So, if you want to change the URL from 'current website name' to a new domain name then  you have to buy another domain and develop website under that domain.

*File address in server:
/domains/yourwebsitename.com/public_html/.htaccess

# BEGIN

RewriteEngine On

RewriteBase /

RewriteRule ^(.*)$ http://yourwebsitename.com/$1 [R=302,L]

# php -- BEGIN cPanel-generated handler, do not edit

# Set the "ea-php72" package as the default "PHP" programming language.

 AddHandler application/x-httpd-ea-php72 .php .php7 .phtml

# php -- END cPanel-generated handler, do not edit

# END

Caution: If the change of domain name becomes unsuccessful you may lose the domain authority and SEO benefits of your old website.