$query = "SELECT * FROM artist WHERE region = '.us.' AND LEFT(english_name, 1) = ".$_GET['letter']."";
Is interpretted as this for example
SELECT * FROM artist WHERE region = '.us.' AND LEFT(english_name, 1) = a
You have no single quotes signifying that a is not a field (column) name.
$query = "SELECT * FROM artist WHERE region = '.us.' AND LEFT(english_name, 1) = '".$_GET['letter']."'";
Try that I added the single quotes in, also I suggest validating $_GET['letter'] as that could be anything, even an SQL Injection Attack (do a good search on what that is).