'Exponential' is a curve shape, not a synonym for 'out of control'.
Linear growth is a different shape - it would mean, in the Covid context, that a constant amount of cases are added in each period of time. If that constant amount is small, then that's less worrying than if the constant amount is huge.
So start with 2 cases and add 2 per day, linear growth is 2,4,6,8 etc
Exponential growth means that, rather than a constant amount being added in a period, the number of cases is multiplied (often quoted as doubled) in a period.
So start with 2 cases and multiply by 2 per day, exponential growth is 2,4,8,16,32 etc
In the case of exponential growth, the factor that matters is the doubling time. If the doubling time is a year, then although growth is still exponential, it's not that worrying. If it is a day, then growth is exponential and very worrying!
So it is a case where both sides of this are right.
The growth is exponential - that describes the shape of the curve. However, the doubling time is lengthening and thus the rate at which cases are growing is slowing. So the growth can be exponential AND flattening out.