What's the easiest way to make them in sync and get turboTax support ? I now have 2 versions of tax preparation in different stage : one via online, another via desktop.

Online and Desktop are separate.  Forget about the Online version and use your Desktop program.  Sorry you can’t sync them.  How did you end up using both versions? 

One thing you can do is download the .tax2023 file from Online for free and open it in the Desktop program.  Be sure to use different names for the files.  Then you can open each return and print them out to compare them.  

Save the tax file from Online

https://ttlc.intuit.com/community/saving/help/how-do-i-save-my-turbotax-online-return-as-a-tax-data-file/00/25740

To continue in the desktop version see this…….

https://ttlc.intuit.com/community/choosing-a-product/help/how-do-i-switch-from-turbotax-online-to-the-turbotax-software/00/26129