What is the reason for the new_to_publish hook not working?

When you create a new post, the initial post status is not new, but rather auto-draft. Try this:

add_action("draft_to_publish", "doSomething", 10,1);

function doSomething($post){

    global $post;

    $_SESSION['yeni'] = 'test';
}

echo $_SESSION['yeni'];

Also note that you may want to make sure your session is properly initialized. To verify the callback works at least, enable WP_DEBUG and WP_DEBUG_LOG and do something like

add_action("draft_to_publish", "doSomething", 10,1);

function doSomething($post){
    error_log( 'Inside the new post transition');
}

Then publish a post and look at wp-content/debug.log