EXPLAIN SELECT id_member, real_name
		FROM forum_members
		WHERE id_group = 1 OR FIND_IN_SET(1, additional_groups)
		LIMIT 33;

*************************** 1. row ***************************
           id: 1
  select_type: SIMPLE
        table: forum_members
         type: ALL
possible_keys: ID_GROUP,debug
          key: NULL
      key_len: NULL
          ref: NULL
         rows: 1369966
        Extra: Using where
1 row in set (0.00 sec)

Better?

EXPLAIN
(SELECT id_member, real_name
FROM forum_members
WHERE id_group = 1)
UNION
(SELECT id_member, real_name
FROM forum_members
WHERE FIND_IN_SET(1, additional_groups))
LIMIT 33\G

*************************** 1. row ***************************
           id: 1
  select_type: PRIMARY
        table: forum_members
         type: ref
possible_keys: ID_GROUP,debug
          key: ID_GROUP
      key_len: 2
          ref: const
         rows: 2
        Extra:
*************************** 2. row ***************************
           id: 2
  select_type: UNION
        table: forum_members
         type: ALL
possible_keys: NULL
          key: NULL
      key_len: NULL
          ref: NULL
         rows: 1375021
        Extra: Using where
*************************** 3. row ***************************
           id: NULL
  select_type: UNION RESULT
        table: <union1,2>
         type: ALL
possible_keys: NULL
          key: NULL
      key_len: NULL
          ref: NULL
         rows: NULL
        Extra:
3 rows in set (0.00 sec)

Though, I must admit I'm not sure an index on additional_groups would help since it's a function. Better here would probably be to normalize it so you have

forum_user_groups
 - user_id
 - group_id