(Mildly). I be thinking a piecewise fit might be a good idea.
Exponential, then fit to parabolic.
That being said — since the disease is NOT randomly distributed across the country, it might be a thought to do the piecewise fit separately for each of (say) a dozen blue city hotspots and go from there; realizing that day 0 for one city might lag day 0 from another.
And mindful that social distancing being unevenly practiced — hardly at all at first, and clamping down as people get alarmed — might affect the classic Gaussian-over-time parabolic shape.
Yeah, I wasn’t trying to say one method was better than another. All I was trying to say was no matter how you fit data, unless the behavior is completely predictable, extrapolating predictions is not reliable. Unfortunately sometimes the message gets lost in egos.