With the release of New Light (Shadowkeep), Forges were simplified extensively. You unlock forges through the New Light main quest, which will lead you to talk to Ada-1. She will give you a Machine gun frame to work on and complete. The Forges themselves are on a per day rotation and accessible from the EDZ map near the Winding Cove landing zone. You don't activate forges by traveling to them directly. Once you have completed the Machine gun, you are done with anything related to unlocking the forges. All weapon frames you acquire from Ada-1 can be completed in any of the forges. If you need a specific forge, you'll need to wait for it to rotate in. The forges are in this order: Volundr -> Gofannon -> Izanami -> Bergusia -> Volundr -> ... For the Izanagi's Burden quest, it should take as little as 4 days to complete provided you acquire a Rare bounty from Ada during that time. These are dropping quite regularly, and I got one on completing my first Weekly bounty from Ada (the ones that reward a Ballistic Log).