SELECT COUNT(*) FROM links WHERE user_id=43
if($count <= 5)
{
//sql insert
}else{
//sql insert
$links_id_arr = ['3', '14', '35', '10', '43' , '23'];
//Находим самый маленкий ИД в $links_id_arr
$min_in_arr = 3;
$sql = "DELETE FROM links WHERE user_id=43 AND id= $min_in_arr";
}