Saving on the web version has always been finicky. I think I have fixed it so that saves won't get deleted from version A.2.1.0 onwards, but I can't guarantee anything.
Cloud Save was supposed to fix the saving issue for the web, but the Cloud Save I use doesn't work with the web version for security reasons. I hope to fix this soon.
For now, I suggest using the downloaded version of the game as it does not have any of these issues (and has Cloud Save).