Growth does NOT have to be exponential (or at least not with any near-constant exponent). Furthermore, if the doubling rate is half a year or more, it really does not matter that much if it exponential, unless the number of already infected is already huge, since a vaccine is likely to come within 2 years.

From my perspective, it appears that politicians do NOT want to take sufficiently drastic measures, as the economic consequences will be huge (let's assume that Spain and Italy will keep or increase the measures they already have for 2 years), and so far, their measures do not seem to be sufficient.