Redirect User when they click Menu Option

The first problem I see is here:

if( get_page_by_slug($slug) ){
}wp_redirect('https://example.com/private-page/'.$slug."https://wordpress.stackexchange.com/");
exit;

The if() statement will do nothing; your wp_redirect() and exit statements are outside the {} braces.

Try this instead:

 if( get_page_by_slug($slug) ){
    wp_redirect('https://example.com/private-page/'.$slug."https://wordpress.stackexchange.com/");
    exit;
 }

Aside from that, a couple other things to check:

• You say the code is in function.php; if you’re referring to a theme file, it needs to be functions.php (note the s) in the currently-active theme.
• Where are these functions being called from? What action/filter hook are you adding them to?