User Login Form Outside the Default wp-login Form

I’m trying to implement a custom registration process consisted of 4 steps.
At the first step user enters her credentials and pushes “continue” to proceed to the next step and the form is sent to the same page.

Here I successfully insert a new user to WP database and I want this user to be logged in right after this. I do it like this:

if ( isset($_POST["user-email"]) && isset($_POST["user-password"]) ) :

// Getting data from the Form
$user_login = esc_attr($_POST["user-email"]);
$user_password = esc_attr($_POST["user-password"]);
$user_email = esc_attr($_POST["user-email"]);

// Composing an array with user credentials data
$user_data =
    array(
        'user_login'    =>  $user_login,
        'user_pass'     =>  $user_password,
        'user_email'    =>  $user_email,
        'role'          =>  'student'
    );

// Inserting new user to the db
wp_insert_user( $user_data );

// Trying to log in
$creds = array();
$creds['user_login'] = $user_login;
$creds['user_password'] = $user_password;
$creds['remember'] = true;

// As far as I know whis does the trick
$user = wp_signon( $creds, false );

if ( is_wp_error( $user ) ) echo $user->get_error_message();

endif;

The problem starts from here. wp_signon for some reason does not login user, which I check by the following:

if ( is_user_logged_in() ) : echo "Yuppiee!";
else : echo "FAIL!";

And I get “FAIL” all the time. Then I dug up the Internets and tried the following:

wp_set_current_user( $user->ID );

if ( is_user_logged_in() ) : echo "Yuppiee!";
else : echo "FAIL!";

And I have my “Yuppiee!” finally, but not for long. After page is reloaded I FAIL! again.

At the end I’ve got a new user added in my WP database and I can see her in the wp-admin/users panel and I can login this user through default wp-login form and I’ve got Yuppiee! all the way. But when I try to login the same user programmatically I got FAIL on page reload.

Can you help me to get through this? Please?!