The Web Server Bundle is not included with Symfony 5.
But you can simply require
it and install it separately.
E.g.:
composer require symfony/web-server-bundle 4.4
It is important that you specify the version (4.4), because otherwise it will attempt to install version 5 (which does not exist, and it will fail).
After that you'll be able to run bin/console server:run
as you used to do.
Otherwise, you may use the Symfony CLI as well. This is an executable binary which includes the Symfony Server by default. Then you may run symfony server:start
or the better know alias symfony serve
to start the local webserver.
Read more about Symfony's server here.
You can also use the built-in web server in the PHP runtime. Just go to your project's root directory and run:
php -S localhost:8000 -t public/
It's not a fully featured webserver, but for developing purposes it is usually more than enough.