Exponential recovery is a little bit misleading as many factors are necessary to achieve just that. The unfortunate thing about any new diseases with a high infection rate such as COVID-19 (R0~2.0,2.2) is that they will grow exponentially under any circumstance but won't assure an exponential recovery rate. To add to that, the "flattening the curve" infographic we usually see is portrayed as the most ideal case.
This video/simulation gives you various scenarios and variables and it neatly shows you how the behaviour of the virus changes: Modeling an epidemic
With the current measures we have now (i.e. social distancing, travel restrictions, and mass testing) we are on track for having more and more recoveries and a higher chance of containing the disease which is good news. You're absolutely right that it may take a long time but that's okay.
Thanks to our hardworking frontline healthcare warriors who's truly battling a noble fight.