The first thing to do is to understand why your health suddenly went so negative. Every city you found costs 4 health with the penalty increasing by ¾ health per population point. Therefore, if you want to maintain 0 or more health at all times, you're going to have to hold off on making new cities until your health is already fairly positive.
The second thing you need to understand is that any buildings or improvements which generate health are generating health local to that city. The total amount of local health in a city is always capped at that city's population. So, even if you're swimming in Energy, and can afford all the +health buildings in the game right from the moment you build a new city, it's still not going to offset its own health cost until its population goes up several times (a city can break even at population 16 but will never go positive).
Therefore, you're going to want to do two things. Firstly, prioritize +health buildings in your larger cities over those same buildings in smaller cities. The effects of health are entirely global, so a new city can easily be offset by several larger, health-filled cities.
Secondly, try to grab some global sources of health. The green virtue tree has some of this (the whopping +7 in the middle row is basically a free new city's worth of health, plus some extra while the 25% less unhealth in the last row can allow several new cities). There are also some wonders which provide health, which I believe count as global.
Of course, there's always an alternative option: just don't pay too much attention to unhealth until it gets particularly bad. The penalties for unhealth aren't too terribly powerful, especially for anything less than -10. And while the bonuses for having positive health are useful, they're hard to achieve and maintain in the early parts of the game. As you get more technology, more options for health, and more large cities with clinics, pharmalabs, biowells, etc., health will become even less of a problem.