PHP SESSION Conflicts - php

Question

I have this in my $_SESSION setting script:
<?php
//----------------------// Start session----------------------
if(!isset($_SESSION))
{
session_start();
}
//------------------------------------------------------------
//------------------// Check if Username $_SESSION is set------------------------------------------
if (!$_SESSION['Username']) { // If not current User
header("Location: ./logout.php"); // Session destroy file that leads to session logout landing page
exit();
}
//------------------------------------------------------------
?>
Now, what I basically do is just check if Username SESSION is set. But, I have come to notice something strange while putting another user through:
If we click the same link at the same time and arrive on the landing page same time, I noticed I can see my Username displayed as his Username and his personal data like email and phone replaced mine in my very own PC! This is really strange to me as we do not even live in the same country or even share same PC.
So, it is obvious I have not secured my SESSION and I have used a lame approach without thinking about security and this can be abused with SESSIONS hijacked.
How do I resolve this conflict? How do I restrict each logged in user to a particular session without conflicts if two or more users access the same resource at the very same time? I need help. I can't sleep since I found this.
After reading your responses, I will now show a snippet of the functions.php file which outputs Use data from DB.
First, I get the UserName value from session using:
$UserName = $_SESSION['Username'];
With this value, I query DB to get more user details:
//------------Get User Info -- All user column
$Get_User_Info = mysqli_query($conn,"SELECT * FROM customers WHERE User='$UserName'");
/************************************************************/
/************************************************************/
$Get_User_Info_row = mysqli_fetch_array($Get_User_Info,MYSQLI_ASSOC);
/************************************************************/
//---- Now list all user rows
$GLOBALS['Skype'] = $Get_User_Info_row['Skype'];
$GLOBALS['Jabber'] = $Get_User_Info_row['Jabber'];
$GLOBALS['ICQ'] = $Get_User_Info_row['ICQ'];
$GLOBALS['Join_Date'] = $Get_User_Info_row['Join_Date'];
$GLOBALS['Join_Date_Time'] = $Get_User_Info_row['Join_Date_Time'];
$GLOBALS['Balance'] = number_format($Get_User_Info_row['Balance'],2);
The above is what is contained in the functions.php which I require with each page I need protected.
As you can see, I barely see where I have done too much wrong there.