It is
not growing geometrically (or only with a tiny multiplier if so). If the feared scenario of exponential growth were the case then we'd see the proportion of positive test cases doubling, not just the raw number. Right now what's being seen is that if you increase testing 10x you find 10x more cases, which is consistent with finding something that's at a somewhat steady background level.
e.g. here's German data:
https://swprs.files.wordpress.com/2020/03/infizierte-pro-tes...
and US data:
https://twitter.com/FScholkmann/status/1246122535793680386
It looks like it doubles every few days because of the rapid increase in testing.
Moreover these hospitals are now entering their second week of being idled. They should be very busy by now if the sick were really growing exponentially - they're still mostly empty.
Even in New York you see this:
https://www.dailymail.co.uk/news/article-8173563/Elmhurst-ta...
"New York City's Elmhurst hospital - the 'epicenter of the epicenter' - is now receiving fewer patients but they are arriving sicker, doctor warns, as he says some come in with no symptoms other than diarrhea then test positive"
"He said testing was surprising and that some people show up with a fever and cough but test negative. Others who are there for different ailments - like car accident victims - end up testing positive."
New York is supposed to be the epicentre of the outbreak yet the most overloaded hospital is now seeing fewer patients arrive than before. That's not consistent with being at the start of a very long exponential growth phase (it obviously can't grow exponentially forever so this discussion is only about how long it lasts in that phase and where the peak is).
Edit: got throttled, will reply to Joe here
Many sources show only the number of positive cases and deaths. Here's one that shows total tests performed in the USA:
https://covidtracking.com/data/us-daily
On the 5th April 2020 there were 332,308 positive cases in the USA. So the halving point was between 29th and 30th March (139,061 and 160,530 cases respectively). It took about 5 days to double.
On the 5th April 2020 there were 1.42 million negative cases. On 29th March 2020 there were 692,290 negatives or 48% of the figure today. It took exactly the same amount of time to double.
So we can see that number of tests doubled in that time. Total tests went from 831,351 (47%) to 1,762,032.
The proportion of positive to negative cases is 18% today. On the 29th March it was 16%. A 2% rise, nothing even close to doubling. The graph in the tweet I linked to shows this visually - a fairly smooth and slow increase over time. We think it's spreading exponentially because of misuse of data, but all that's actually growing exponentially is testing. And yes - that's probably why there are now global shortages of reagents and other ingredients for tests. You can't keep globally doubling demand for tests without eventually hitting production limits!