I agree with these concerns. I'm using the Android version and some event cards will stay displayed on the screen briefly, but not all of them. Some then seem to stay for a while after you've resolved their actions. This becomes very annoying to have to go back and forth to the game log and rules link (which some don't have links and most don't show actual card instructions, just requirements) when it would seem easy to leave a small icon of the card off to the side until the end of the turn that could be clicked on to enlarge like the other cards.
If possible, this would be a very helpful feature to add to the next update.