Generally the Arrowverse shows take place in roughly airdate order, but that doesn't make sense for Batwoman and I'm having trouble placing it.
In "Elseworlds", Batwoman has a red wig and full bat suit, which she doesn't get until Batwoman episode 3. She doesn't even know who Batman is prior to Batwoman episode 1.
There is some reality-rewriting in "Elseworlds", but Kate recognizes Kara in "Crisis of Infinite Earths Part One", so her part of "Elseworlds" doesn't appear to have changed.
Batwoman has had only eight episodes so far, and there don't appear to be any long timeskips between episodes, but Kara refers to the events of "Elseworlds" as "last year" at the end of "Crisis of Infinite Earths Part Two".
Prior to "Crisis of Infinite Earths" I thought the entire first season of Batwoman would take place before "Elseworlds", but at the end of "Crisis of Infinite Earths Part Two", Kate refers to the events of Batwoman episode 8 as if they were recent.
Is there any way of placing the episodes of Batwoman in the rest of the Arrowverse timeline that makes sense?