Recursive self relational via join column
Here is an example query,

WITH RECURSIVE t(item_id, json) AS (
        SELECT item_id, to_jsonb(items)
        FROM items
        WHERE NOT EXISTS (
                SELECT 1
                FROM joins
                WHERE items.item_id = joins.item_id
        )
        UNION ALL
        SELECT parent.item_id, to_jsonb(parent) || jsonb_build_object( 'children', t.json )
        FROM t
        JOIN joins AS j
                ON t.item_id = j.child_id
        JOIN items AS parent
                ON j.item_id = parent.item_id
)
SELECT item_id, jsonb_pretty(json)
FROM t
WHERE item_id = 1;
 item_id |             jsonb_pretty              
---------+---------------------------------------
       1 | {                                    +
         |     "title": "PARENT",               +
         |     "item_id": 1,                    +
         |     "children": {                    +
         |         "title": "LEVEL 2",          +
         |         "item_id": 2,                +
         |         "children": {                +
         |             "title": "LEVEL 3.2",    +
         |             "item_id": 6             +
         |         }                            +
         |     }                                +
         | }
       1 | {                                    +
         |     "title": "PARENT",               +
         |     "item_id": 1,                    +
         |     "children": {                    +
         |         "title": "LEVEL 2",          +
         |         "item_id": 2,                +
         |         "children": {                +
         |             "title": "LEVEL 3.1",    +
         |             "item_id": 3,            +
         |             "children": {            +
         |                 "title": "LEVEL 4.1",+
         |                 "item_id": 4         +
         |             }                        +
         |         }                            +
         |     }                                +
         | }
       1 | {                                    +
         |     "title": "PARENT",               +
         |     "item_id": 1,                    +
         |     "children": {                    +
         |         "title": "LEVEL 2",          +
         |         "item_id": 2,                +
         |         "children": {                +
         |             "title": "LEVEL 3.1",    +
         |             "item_id": 3,            +
         |             "children": {            +
         |                 "title": "LEVEL 4.2",+
         |                 "item_id": 5         +
         |             }                        +
         |         }                            +
         |     }                                +
         | }
(3 rows)


Note, we're not actually merging the paths to form a completed tree. You either have to build the tree from the root node down, or the leaf nodes to the top. In this case, you'll have to merge the discrete paths. Look for a deep json merge in Javascript and tie it together with plv8

Test data

I modified your example schema a bit,

DROP TABLE joins;
CREATE TABLE items (
  item_id     serial PRIMARY KEY,
  title text
);
CREATE TABLE joins (
  id          serial PRIMARY KEY,
  item_id     int,
  child_id    int
);
INSERT INTO items (item_id,title) VALUES
  (1,'PARENT'),
  (2,'LEVEL 2'),
  (3,'LEVEL 3.1'),
  (4,'LEVEL 4.1'),
  (5,'LEVEL 4.2'),
  (6,'LEVEL 3.2');
INSERT INTO joins (item_id, child_id) VALUES
  (1,2),
  (2,3),
  (3,4),
  (3,5),
  (2,6);