Add logo in navigation bar before menu items in twenty thirteen?

Tested on a Twenty Thirteen & Twenty Twelve child theme.

enter image description here

Try this from your child themes functions.php file.

You can add the image using various methods.

add_filter( 'wp_nav_menu_items', 'wpsites_add_logo_nav_menu', 10, 2 );

function wpsites_add_logo_nav_menu( $menu, stdClass $args ){

if ( 'primary' != $args->theme_location )
    return $menu;


$menu .= '<nav class="nav-image"><img src="' . get_stylesheet_directory_uri() . '/images/header.png" /></nav>';

return $menu;

}

Sample CSS for your child themes style.css file:

.nav-logo {
float: left;
margin-right: 20px;
}

You’d also want some Media Queries which will vary per theme.

Here’s the result on Twenty Twelve:

enter image description here