What @LoicTheAztec is one good solution but if you want another solution you can use the following:

WordPress have built in function which allow you to remove the default callback function for your shortcode and replaced it with your custom one.

In this case we are going to check if the cart content count is more than 4 then remove the default callback and replace it with ours .

i will consider for example that the page id which you have is 49 here which you should change when using this code to match the page which contain the shortcode.

//Our Check 
function checkShortCode()
{
$page = get_post(49);
if (WC()->cart) {
$items_count = WC()->cart->get_cart_contents_count();
if ($items_count == 4) {
//Remove the Default Hook function for this shortcode
remove_shortcode(\'wof_wheel\');
//Add custom callback for that short to display whatever message you want
add_shortcode(\'wof_wheel\', \'myCustomCallBack\');
}
}
}
add_action(\'wp_loaded\', \'checkShortCode\');

now we need to add our custom callback to display whatever message you want:

function myCustomCallBack()
{
echo \'my shortcode is running\';
}

the code above is tested and working 100%

You can add conditional shortcode by using below code :

    $cartcount = WC()->cart->get_cart_contents_count();
if ($cartcount > 4) { 
echo do_shortcode( \'[wof_wheel id=\\\"1111\\\"]\' );
}else{
//item count is less than 4
}

Note : In the above question you re-declare the predefined function which is incorrect , do_shortcode() is predefined function for echo the shortcode in template file.

For more help see this link : Click Here

You can embed a shortcode in a custom shortcode with the needed conditional logic:

add_shortcode( \'my_wheel\', \'custom_conditional_wof_wheel\' );
function custom_conditional_wof_wheel( $atts ){
$atts = shortcode_atts( array(
\'id\' => \'\',
\'count\' => \'4\', // 4 cart items by default
), $atts, \'my_wheel\' );
// If there is more than 4 items count in cart the shortcode [wof_wheel] is executed
if( WC()->cart->get_cart_contents_count() > $atts[\'count\'] ){
$id = $atts[\'id\'];
return do_shortcode( \\\"[wof_wheel id=\'$id\']\\\" );
}
// Else it display nothing
return \'\'; 
}

Code goes in function.php file of your active child theme (active theme). tested and works.
(see at the end how I have tested that).

USAGE:

1) For more than 4 items in cart (4 items is set by default in the shortcode):

[my_wheel id=\\\"1111\\\"]

2) For more than 6 items in cart for example:

[my_wheel id=\\\"1111\\\" count=\'6\']

How this has been tested.

As I can\'t test this shortcode that comes from a specific third party plugin, I have created a [my_wheel] shortcode that will output the ID provided in the shortcode argument id:

add_shortcode( \'my_wheel\', \'custom_conditional_wof_wheel\' );
function custom_conditional_wof_wheel( $atts ){
$atts = shortcode_atts( array(
\'id\' => \'\',
\'count\' => 4,
), $atts, \'my_wheel\' );
if( WC()->cart->get_cart_contents_count() > $atts[\'count\'] ){
$id = $atts[\'id\'];
return do_shortcode( \\\"[wof_wheel id=\'$id\']\\\" );
}
return \'\';
}

Code goes in function.php file of your active child theme (active theme).

Then I have added [my_wheel id=\\\"1111\\\"] shortcode in the Wordpress text editor of a page and I get this display when the cart items count is 5 or more:

So it works.