Do they say you did not file a return or that you did not pay your tax due? Your only proof that you paid tax you owed would be your own bank/credit card records.
What does it say in your account? Does it show that your e-filed return was accepted? If the state cannot find it, then print it, sign it, and mail it to them. If you still owe tax, you will have to pay it.
See this TurboTax support FAQ on how to contact Support - https://ttlc.intuit.com/questions/3401489
Use this website to contact TurboTax support during business hours - https://support.turbotax.intuit.com/contact/
Use this phone number and select TurboTax - 1-800-4-INTUIT (1-800-446-8848)
On every TurboTax web page, including this one, scroll down to the bottom of the page and click on Contact Us
Phone support is not provided with the Free Edition. If you are using a paid version of the software or if you purchased PLUS you can get phone support when customer support is there.
To call TurboTax customer support
They are available from 5 a.m. to 5 p.m. Pacific time Monday-Friday
You can look in your account to see if the return shows as having been accepted or rejected.
When you e-file your federal return you will receive two emails from TurboTax. The first one will say that your return was submitted. The second email will tell you if your federal return was accepted or rejected. If you e-filed a state return, there will be a third email to tell you if the state accepted or rejected your state return.
You first need to mitigate your damages. Print out, sign, and mail a copy of your state return to the FTB as soon as possible. Use mail with tracking if you can. Turbotax would not be responsible for extra damages caused by your delay in responding to the state.
The Turbotax Accuracy Guarantee would seem to apply here -- you always owe the correct tax you owe, but if the state assesses penalties or late fees due to a Turbotax error, the accuracy guarantee is supposed to cover it. You must file a claim within 60 days of getting notice of a penalty assessment.
If you have not been assessed a penalty and are just grumpy, you can try calling customer support. If you want something concrete, like a refund, you can ask. If you want an explanation, I'm not sure what is going to help you -- if Turbotax says it was transmitted and accepted and the FTB says it was never received, how do you know the FTB didn't lose it?
(Did you get two separate acceptance notices, one from the IRS and one from the state? Maybe the IRS accepted it but the state rejected it and you misread the acceptance notice from the IRS.)