They are not trying to "soak" you for more. They are trying to get you to pay the correct amount of tax on your income.
Form 4963 means you have a balance due. How much does the notice say you owe? If it's the amount that was due on your tax return, then either they did not get your payment or they did not properly credit it to your tax due. How did you pay the tax? Your proof of payment is a canceled check, a bank record showing the payment, or a credit card statement showing the payment. If the payment did come out of your bank account or credit card account, follow the instructions in the notice for what to do if you disagree with it.
As far as having to pay instead of getting a refund, it's not "a TurboTax thing," it's a taxpayer thing. Assuming you were employed, you apparently did not have enough tax withheld from your pay during the year. You can have your employer withhold more tax so that you will get refund. For your federal tax you can use the Tax Withholding Estimator on the IRS web site to figure out what to put on the W-4 form that you give to your employer. It will show you how to adjust your withholding to get the approximate refund amount that you want. For California use Form DE 4.
But keep in mind that it's better to owe a small amount when you file your taxes instead of getting a refund. Getting a refund means that you paid more tax than necessary during the year. The IRS or FTB holds your money and sends some of your own money back to you when you file your tax return. And they don't pay you interest on the money that they held. If you adjust your withholding so that you owe a small amount at the end of the year you get to use the money during the year instead of having the government hold it. And you don't have to wait for the refund or worry about the refund being delayed or lost.
