You're doing an echo and then writing a relocate header. If you did your relocate in the javascript (after the user clicked the alert), it would probably work the way you expect it to.

echo "<script>alert('You are logged out'); window.location.href='..\Main.php';</script>";

Also, the way that you use isset will cause problems because isset returns true or false (it checks if a value is present), rather than returning the value.

So instead of

if(isset($_SESSION['Username']) == "admin")

You need to do:

if(isset($_SESSION['Username']) && $_SESSION['Username'] == "admin") 

It's easy: you have Location header:

 echo "<script>alert('Addition Successfull !!');</script>";
 header("Location: banking.php");

When you send Location header to user, he/she will get redirected and script won't be interpreted!

P.S: As @Quentin said, mysql_ functions are deprecated, they're old and no longer maintained. Use mysqli or PDO.

P.S 2: You send content before header, so assume you're using output buffering.

It would be better if you use below mentioned code.

if(empty($response)){ 
//SOME ACTIONS
?>
 <script type="text/javascript"> alert("Message´s 1st line\nMessage´s 2nd line");</script>
<?php
//SOME ACTIONS
header('Location: SOMEWEBSITE');                                
}else{ 
//ANOTHER ACTION
} 

You are not using any response in your ajax request. you need to get response as:

success: function(response){

Don't know why are you using die() here, you can just use echo here as:

if($data_exists){
    echo 'The person exist';
}else{
    // else part
}

But, still how can you check either getting success or error in ajax response. you can use json here like:

if($data_exists){
    echo json_encode(array('status'=>0,'message'=>'The person exist'));    // for error 
}else{
    // else part
    echo json_encode(array('status'=>1,'message'=>'Success message')); // for success
}

After this, you need to change your ajax as:

$.ajax({
    type:'POST',
    url:url,
    data:$('#formulario').serialize(),
    dataType: "json",  // will return json response
    success: function(response){
        if(response.status){
            // your success part
        }
        else{
            alert(response.message); // error response in alert
        }
    }
});

Other Improvements:

You can use Prepared Statement for preventing your SQL Statements with SQL Injection.

Your close button is attempting to fire a JavaScript function called killIt() with a parameter of 'error'. I'm going to guess that you haven't included that function on the page, or that there's an error in it.

Using header("Location: ...") will cause PHP to respond with a 302 Found status, rather than a 200 Success status.

A 3xx status will cause the browser to do a redirect, ignoring anything you send in the body (including JavaScript).

Instead you can alert and then redirect using JavaScript.

} else {
  unset($_SESSION['insertQuery']);
  echo "<script>alert('Success'); window.location.href = 'bluk_resource_booking.php?successInsert=1';</script>";
  exit;
}

use this code

echo '<script language="javascript">';
echo 'alert("message successfully sent")';
echo '</script>';

The problem was:

1. you missed "
2. It should be alert not alery