I'm curious to see how I could waive part of my health insurance penalty, especially for starting a new job in December 2021, and being in a probationary period but starting a health insurance plan once out of that period. Is there a situation or category to select in TurboTax that accounts for this? (And yes, I was already working this job in my 2021 taxes but was not a fully covered employee yet)
There is no federal tax penalty for tax year 2021 or 2022 for not having health insurance, so I believe you are referring to a state tax penalty. If so, you need to tell us what state you live in please.
Among other exemptions allowed by California, there's a "Short coverage gap of three consecutive months or less" it appears that you may qualify for.
Please see Exemptions processed by FTB and Covered California on the Franchise Tax Board's Personal Health care mandate page for more information.