I'm doing something like that.
Getting the values in lowercase and MySQL does the rest
$string = $_GET['string'];
mysqli_query($con,"SELECT *
FROM table_name
WHERE LOWER(column_name)
LIKE LOWER('%$string%')");
And For MySQL PDO Alternative:
$string = $_GET['string'];
$q = "SELECT *
FROM table_name
WHERE LOWER(column_name)
LIKE LOWER(?);";
$query = $dbConnection->prepare($q);
$query->bindValue(1, "%$string%", PDO::PARAM_STR);
$query->execute();