If you purchased the TurboTax software from the TurboTax website then sign onto your account and download the software again - https://shopping.turbotax.intuit.com/unified/downloads?route=myaccounts&cgcview=downloads
How did you install it on the first computer? By CD or by using the download? I know you said you downloaded it but a lot of times people mean from the CD. If you actually used the Download method you can download it again the same way. Did you get it from Turbo Tax or somewhere else like Amazon or Costco? Do you still have the License Code?
How to download your purchase
https://ttlc.intuit.com/community/installing/help/how-do-i-download-the-software-i-purchased-at-turbotax-com/00/26075
Access your downloads here
https://shop.turbotax.intuit.com/commerce/account/secure/login.jsp?pagecontext=download
If you can't find your download, use this TurboTax account recovery website to get a list of user ID's for an email address. Run the tool against any email addresses you may have used
https://myturbotax.intuit.com/account-recovery
Did you start your returns? Do you have the files from the old computer? Like a backup copy? If you have the .tax2021 file then after you get the program reinstalled the first thing to do before you open your tax return is to update the program and install any state programs you had. Then open your file. So you first might need to start a fake return to be able to download the state program (go to FILE - NEW). Enter some basic information, then go to the state tab and download the state program. Then try to open your existing return, go to FILE - OPEN