This should work:
$str = 'The quick brown <b>fox</b> jumped <b>over the lazy <b>dog</b>.';
$str_arr = explode('<b>', $str);
foreach ($str_arr as $key => $value) {
if (strpos($value, '</b>')) {
$str_arr[$key] = '<b>' . $value;
}
}
$str = implode('', $str_arr);
echo $str;