First try restarting the Mac.
If it will just not open after installing.
if you get the message "TurboTax cannot be opened because Apple cannot check it for malicious software" then that is caused by the Apple "Gatekeeper" introduced with OS X 10.15 (Catalina).
That has nothing to do with TurboTax and will happen for any application not purchased from the Apple Store.
See this Apple article on this: https://support.apple.com/en-us/HT202491
And this TurboTax help article: https://ttlc.intuit.com/community/troubleshooting/help/turbotax-for-mac-won-t-open-when-installed/00/26611
For older Mac systems when TurboTax will not open.
See:
Be sure that System Preferences -> Security & Privacy -> General -> Allow apps downloaded from App Store and identified developers is checked.
Go to the Applications folder and select the TurboTax application in the Finder and right click with the mouse (or control click) ans select the top "open". That should force the first open. After that it should open normally.
https://ttlc.intuit.com/questions/1899267-administrator-rights-mac
Corruption could have occurred in the Application or in an associated preference or support file.
The best remedy is to totally uninstall TurboTax and all associate files and then re-install. That will not delete or change any .tax2019 data file that had already been created with the tax data.
To uninstall the application and all associated files need to be deleted and then the application reinstalled. If there was a state then the state must also be reinstalled.
Here is a FAQ for how to uninstall Mac
https://ttlc.intuit.com/questions/1901210-remove-uninstall-turbotax-for-mac
https://ttlc.intuit.com/questions/1901209-reinstall-turbotax-mac