Stuck and can't get to Lothric Castle

Sorry in advance if this is not the correct sub for asking for tips. I've completed Irithyll Dungeon, defeated Pontiff Sullyvhan in Irithyll of the Boreal Valley, defeated Aldrich in Anor Londo, and beat Yhorm the Giant in Profaned Capitol.

I thought I'd get automatically transported to Lothric Castle after beating Yhorm or Aldrich but nothing has happened. The only thing I can think that I missed was that I didn't keep up with Siegward's storyline, but I thought that was side quest that didn't affect the main storyline. I helped him out in Undead Settlement but didn't know he was in the well outside Cathedral of the Deep, and he's no longer there. Any ideas what I'm missing to progress the story?