No.   The Apple supplied Developers tools needed to create the new 2020 software do not support 10.13 because Apple chooses not to.   That is a Apple decision.

What version is needed on my iMac?  And, I made the mistake of buying and downloading TurboTax before knowing that, so can I get my money back?

See my answer above.

So I can use High Sierra version 10.14, if there is one.  If not, can I get my money back?

TurboTax requires OSX 10.14 (Catalina) or later.  High Sierra is 10.13.

Of course that is for the CD/Download desktop version installed on your Mac.  Online versions will work with ANY comparable browser and the OS is not important.

How do I get my money back, if I can't upgrade?

Where is the file kept if I do it online?  NOT big on having it in the "Cloud".

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@cluidens wrote:

Where is the file kept if I do it online?  NOT big on having it in the "Cloud".

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On the TurboTax server, but you can download a copy.

Online also does not have all the same features - there is  no "forms mode" to view the forms and each account is only good for one tax return.