Yes, TurboTax supports it. It's in the forms availability list for California. The form number on the form, and in the TurboTax list, is just 3800, not FTB3800. But the California instructions refer to it as FTB 3800.
See https://www.ftb.ca.gov/individuals/FileRtn/7.shtml, which says, in part:
For taxable years beginning on or after January 1, 2010, California law conforms to federal law which allows parents’ election to report a child’s interest and dividend income from children under age 19 or a student under age 24 on their tax return. For each child under age 19 or student under age 24 who received more than $2,100 of investment income in 2018, complete Form 540 and Form FTB 3800, Tax Computation for Certain Children with Investment Income, to figure the tax on a separate Form 540 for your child.
