Exponential growth, in this context, means that the rate of new infections is proportional to the number of existing infections. This cuts both ways: when the number of existing infections is large, the rate of growth is also large -- which is why exponential growth is so dangerous -- but when the existing number is small, the growth rate is also small. If you look at a graph of an exponential curve, there's a long relatively flat region to the left, before the knee in the curve is hit and the thing takes off.
The point is, even though we here in CA started the lockdown relatively early, we've never gotten the case numbers down to a manageable level. I'm not sure why not. Although a lot of us have taken the virus seriously from the beginning, maybe there are still too many who don't. The sad truth is that a lot of people won't believe it until it affects someone they know -- and while it's well under control, that tends not to happen. So it looks like a bad outbreak in a region creates "mental herd immunity", where almost everyone cooperates with the lockdown; places that haven't had such an outbreak don't take it as seriously.