Rather than including the two club objects inside of the clubs array, you only adding the Oak Club object to it and then defining another array to the JSON object that is anonymous. Here's a better way to structure that code:

json_clubs = {
    "clubs": [
        {
            "Oak Club": {
                "members": [
                    "Charles",
                    "Nora",
                    "Tom",
                    "Paul"
                ]
            }
        },
        {
            "Bravo Club": {
                "members": [
                    "Nathaniel",
                    "Roselyn",
                    "Ash"
                ]
            }
        }
    ]
}

As we found in the comments that there are invisible characters in the string, consider running this bit of code. It should show you each character in the string, it'd ascii value and it's printed version.

There should be a few characters in there which do not print anything; those are the ones preventing you from json_encoding the string. I cannot see why they are in the string, that has something to do with your database and the data you post into it, but at least you'll know where the problem is.

Odds are they are contained inside the inserted data somewhere, which means you probably have a piece of code somewhere that add something unprintable to their input.

You can strip them while exporting, but it would be better to find out which code adds them and remove them from the database. They will continue to bother you in the future otherwise. (As you've seen at this point even something simple like strlen() will fail to work properly due to these characters)

for( $i = 0 ; $i < mb_strlen($string) ; $i++ ) {
  echo 'pos: ' . $i . ' | ord: ' . ord( $string[$i] ) . ' | char: ' . $string[$i] . '<br />';
}

The fact the character is a < make me think you have a PHP error, have you tried echoing all errors.

Since I don't have your database, I'm going through your code trying to find errors, so far, I've updated your JS file

$("#register-form").submit(function (event) {

    var entrance = $(this).find('input[name="IsValid"]').val();
    var password = $(this).find('input[name="objPassword"]').val();
    var namesurname = $(this).find('input[name="objNameSurname"]').val();
    var email = $(this).find('input[name="objEmail"]').val();
    var gsm = $(this).find('input[name="objGsm"]').val();
    var adres = $(this).find('input[name="objAddress"]').val();
    var termsOk = $(this).find('input[name="objAcceptTerms"]').val();

    var formURL = $(this).attr("action");


    if (request) {
        request.abort(); // cancel if any process on pending
    }

    var postData = {
        "objAskGrant": entrance,
        "objPass": password,
        "objNameSurname": namesurname,
        "objEmail": email,
        "objGsm": parseInt(gsm),
        "objAdres": adres,
        "objTerms": termsOk
    };

    $.post(formURL,postData,function(data,status){
        console.log("Data: " + data + "\nStatus: " + status);
    });

    event.preventDefault();
});

PHP Edit:

 if (isset($_POST)) {

    $fValid = clear($_POST['objAskGrant']);
    $fTerms = clear($_POST['objTerms']);

    if ($fValid) {
        $fPass = clear($_POST['objPass']);
        $fNameSurname = clear($_POST['objNameSurname']);
        $fMail = clear($_POST['objEmail']);
        $fGsm = clear(int($_POST['objGsm']));
        $fAddress = clear($_POST['objAdres']);
        $UserIpAddress = "hidden";
        $UserCityLocation = "hidden";
        $UserCountry = "hidden";

        $DateTime = new DateTime();
        $result = $date->format('d-m-Y-H:i:s');
        $krr = explode('-', $result);
        $resultDateTime = implode("", $krr);

        $data = array('error' => 'Yükleme Sırasında Hata Oluştu');

        $kayit = "INSERT INTO tbl_Records(UserNameSurname, UserMail, UserGsm, UserAddress, DateAdded, UserIp, UserCityLocation, UserCountry, IsChecked, GivenPasscode) VALUES ('$fNameSurname', '$fMail', '$fGsm', '$fAddress', '$resultDateTime', '$UserIpAddress', '$UserCityLocation', '$UserCountry', '$fTerms', '$fPass')";
        $retval = mysql_query( $kayit, $conn ); // Update with you connection details
            if ($retval) {
                $data = array('success' => 'Register Completed', 'postData' => $_POST);
            }

        } // valid ends
    }echo json_encode($data);

Removing the BOM (Byte Order Mark) is often-times the solution you need:

function removeBOM($data) {
    if (0 === strpos(bin2hex($data), 'efbbbf')) {
       return substr($data, 3);
    }
    return $data;
}

You shouldn't have a BOM, but if it's there, it is invisible so you won't see it!!

see W3C on BOM's in HTML

use BOM Cleaner if you have lot's of files to fix.

JSON.parse("[{\"name\":\"gath\",\"age\":10}]");

You cant have double quotes inside double quotes

The data is not valid JSON, since \" in strings seems to have been replaced with "\.

Contact the author of that supposedly-JSON and notify him or her that there are plenty of JSON libraries available for all languages and platforms.