To call TurboTax customer support

https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number

They are available from 5 a.m. to 5 p.m. Pacific time  Monday-Friday

Does your Mac meet the system requirements for the 2020 software?

Mac CD/Download Products

Operating Systems

• macOS Mojave 10.14 or later

RAM

• 2 GB or more recommended

Hard Disk Space

• 1 GB for TurboTax

Monitor

• 1024x768 or higher recommended

Internet Connection

• 1 Mbps modem (Broadband connection highly recommended). Required for product activation, software updates and optional online features

Printer

• Any Macintosh-compatible inkjet or laser printer

Operating Systems

• macOS Mojave 10.14 or later

No I have macOS High Sierra 10.13.6

I am having the same problem, installed and it would not start up.  Then removed it, reinstalled it, it will still not start up.

It will not start up, this is TurboTax 2020 Deluxe, Federal Only with e-File, downloaded from Amazon.

macOS Big Sur 11.1, 8GB RAM

Same here. Tried with Mojave AND Catalina. No success. After install, first time it goes through its "verify" process, then warns me that Mojave is ok this year but not next year, then the TT icon bounces a few times, stops bouncing, and TT is not running. If I try to run TT again, it just bounces a couple of times, then stops. Tried the CD version and the download version. Found one posting that explained how to clear out the Library of TT stuff. Did that. Same problem. This is infuriating!!!!