How do I file back taxes if I legally changed my name in 2023, but still have 2021 and 2022 to complete as well?

Use the name that matches your current Social Security card.   The SSN is the same, right?    Since the past year returns have to be filed by mail, there will be no problem with e-files being rejected.  When it asks for a 2023 AGI to e-file 2024, you will have to use zero since you are filing those past years late.

You cannot change the tax year.   The current online program is for 2024 only.   Only a 2024 return can be prepared online and only a 2024 return can be e-filed.

Online preparation and e-filing for 2021, 2022, and  2023 is permanently closed.

Note:  The desktop software you need to prepare the prior year return must be installed/downloaded to a full PC or Mac.  It cannot be used on a mobile device.

To file a return for a prior tax year

 If you need to prepare a return for 2021, 2022, or 2023  you can purchase and download desktop software to do it, then print, sign,  and mail the return(s)

https://turbotax.intuit.com/personal-taxes/past-years-products/

You may also want to explore purchasing the software from various retailers such as Amazon, Costco, Best Buy, Walmart, Sam’s, etc.

Remember to prepare your state return as well—if you live in a state that has a state income tax.

https://ttlc.intuit.com/turbotax-support/en-us/help-article/state-taxes/contact-state-department-revenue/L9qVToi02_US_en_US?uid=m6e06um0

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.  Use a mailing service that will track it, such as certified mail so you will know the IRS/state received the return.

Federal and state returns must be in separate envelopes and they are mailed to different addresses.  Read the mailing instructions that print with your tax return carefully so you mail them to the right addresses.