This exact thing keeps happening to me. The brief pop up claiming, "checking to make sure everything is up to date" and then shutting down immediately. I was able to download the software originally, complete my tax forms to the very end - I had even paid for the state return, which turbo tax happily charged my credit card, and then it crashed, never to be opened again. I'm just desperate to finish them.
I'm on Mojave, so it's not just a Catalina problem. But I'm afraid to update now.
