How to increase colony's health

what can I do to prevent these extreme dips of health in the future and what can I do to increase it both short and long-term?

I started off like this - worrying about health, but I think the answer is: Completely ignore it - if you want more than two cities, you're going to be unhealthy for long periods of the game pretty much no matter what you do.

The penalties are not bad - the negative science/culture is more than made up for the fact that you can then build as many cities as you want, as fast as you want (the slowed city however is annoying). It's also worth noting that -2000 health is no worse than -20 health. Once you start down the unhealthy lots-of-cities path, there's almost no point in even building health improvements as you're likely to be so far in the red that it makes no difference. In the late game where you get cultural advances that can offset the unhealthiness, you can then do something about building health improvements in cities.

In the prosperity tree in the virtues, there is the last virtue of the second tier wich gives you +7 ( its called mind over matter ). There are other health bonuses in the virtue's, like +1 health every 6 citizen, 0.25 health for every military unit,... Other than that, focus on the researches witch eventually give you health through resources or buildings. It is normal to be on a negative health state at some point of the game, its very though to always stay in the + side.

Really good health virtue is in industry tree. It gives 0.5 health per trade unit. You have 2 trade routes per city (and with autofactory it is 3 (other choice is extremely weak). This gives 1.5 H per city early.