Choosing the right template
As your “User Profile”-Page is something completely custom and not the admin UI user profile page, I’d suggest to take the uthor posts page instead:
get_author_link( true, get_current_user_id() );
Then modify this template.
Pretty URls
No need to go outside the WP template hierarchy. If you want something like ~/user, then use the Rewrite API.
( Update )
get_author_link() This function has been deprecated. Use get_author_posts_url() instead.
Example
get_author_posts_url( get_current_user_id() );
https://codex.wordpress.org/Function_Reference/get_author_posts_url