That's what the math says should happen. It's not what the measurements say. If you look on http://91-divoc.com/pages/covid-visualization/ you will see country after country goes linear. The USA for example had 30k new infections most day for a remarkable 2 months.
I have no idea what causes it. It could be measurement error (that's what I first put it down to in the USA). But if it is it damned hard to explain why it has happened over and over again. Russia has been sitting on 9k new cases per day for 2 months now. The UK went through a period of 5k new cases per day for a month.
It's not a universal, or even the most common pattern. But it happens enough to make the statement "the measures of new covoid 19 cases always change exponentially" wrong enough to be effectively useless as a prediction of where things will head.
However, rising %positive and higher reported rates together or rapidly rising hospitalizations (eg. rising even 5-10% per day) are indications that by the time any new (present day) controls have effect on measured outcome (20-30 day delay) the situation will be 5-10x worse.
I agree that CA (especially NorCal) is probably over conservative... but only because there is so much travel that it doesn't matter if they get to zero, when a few travelers form Arizona, Nevada, or LA (ignoring air travel) will bring it all back in a jiffy. What is likely to be the real boom is when kids go back to school... most of them won't get too sick, but most of them have parents or teachers who will. That's your real second wave when everyone is already tired of caring.
Remember, a guy can have unprotected sex with a lot of women before anyone notices they're pregnant. It's probably 3 weeks from infection to ICU and 5 or 6 weeks to death so looking at deaths or even hospitalizations only gives you informations so delayed it's useless when the doubling time can be 3day (and even when it's 10 days). You're up 10x before you know it...