So normally a single district gets pillaged, maybe a couple of mines. In these cases the number of turns is low, if you lose more and all of your mines then a worker or two gets you most of the way there.
Early on when you only have a small border, and a mine or 2 and no surviving support district like an IZ/power station then it can be quite different and worse.
In the worst case later in the game you may lose 1/10 of your entire production, but early game it’s a lot larger.
IRL, early cities disappeared right and left, due to warfare, natural disaster (drought was a big one) and other causes, but in the game, where in the Ancient Era you have, at most, 3 - 5 cities, losing one at near random is Not Acceptable, so the game makes it really hard to lose a city except to major enemy invasion: barbarians, disasters, simply won't do it.
Having 'nerfed' Bad Things To Cities, though, they have maintained Really Bad Things For Districts, which seems to be quite the opposite to the general intent of the game design. If the city is still there but Unproductive, it's almost as bad as Not There, and if its placement on the map antagonizes another Civ, it may be worse.
At the very least, then, Districts should be much easier to repair, or the repair costs more in line with the Resources available to the cities, in the first 2 - 3 Eras of the game. The fact that a city with Pop 15 or more and developed Production Resources can repair a District in a dozen turns does the player no good when the same amount of damage has been done to the district in a city of Pop 6 with a single Watermill!