Shouldn't the child tax credit now be either $3000 or $3600, instead of the $2000 shown on Turbo Taxes software?

No. 

That CTC increase is not for 2020;  it is for 2021.   Your 2020 return does not have to be amended for it.  More information will be available soon; the details of how it will be implemented have not yet been clarified fully.  There may be monthly payments going out beginning in July 2021, with some of the CTC still left as a refundable credit on your 2021 tax return next year.

https://ttlc.intuit.com/community/tax-topics/help/how-will-the-stimulus-package-impact-me/01/1393859#U1393859

No. The $2000 child tax credit given for eligible children in the TurboTax software is correct for Tax Year 2020. The $3000 (or $3600) child tax credit does not take effect until this year, Tax Year 2021. Some of that additional credit will be made in advance payments during the 2021 tax year and any remaining amount owed will be credited when your 2021 tax return is filed in 2022.