It is not because of the corruption but because the program chooses only tiles with food. Focusing on Production on the Governor screen does not make increase it.
The governor will only assign a newborn citizen to a shield-rich tile, if the town is making
at least 3-4 food net before growth, and will still net at least +2 food even when the shield-rich tile is being worked after growth (or when the town reaches a population-growth cap, e.g. is not next to freshwater, so cannot grow beyond 6 citizens until an Aqueduct has been built).
To a great extent, the terrain around your town/ capital will dictate what improvements you will need to add to it. Since each citizen needs 2 food per turn (or they starve), as a general rule, you should aim to get at least 2 food from each worked tile
on average (e.g. if you
have 4 citizens in total, working 4 tiles, you should be getting 8 food from those tiles in total), so that each town is producing
at least 2 extra food
(from the town-tile itself) beyond what your citizens are currently eating, in order to continue growing (as large as possible!).
So e.g. when your capital is surrounded by mostly-Grassland, each of which already gives
at least the net 2 food needed, in the early game, you should usually mine (almost*) every Grassland tile.
(*Tiles with Cows or Wheat might be better irrigated -- if possible -- since that will give you some extra food, i.e. faster growth, even under the "Despotism penalty")
For a Plains start, you
really want a freshwater source (a river or a "lake" = any enclosed body of water less than 21 tiles in area) to use for irrigation (bringing the tile from 1 food to 2 food), or, yes, your town-growth will stall/starve at 3 citizens (but you should also be getting at least 2-3 shields per turn at that point: enough to build a Settler in 10-15 turns). If there is no freshwater nearby, consider simply building lots of small towns, packed closely together ("infinite city sprawl", City-tile-City-tile-City). Later on, if you find freshwater further away, you can 'chain-irrigate' from there back to your capital, and then start removing every second town (e.g. turning those citizens into Workers or Settlers), so you have "City-tile-tile-tile-City".
Floodplain-starts tend to be very food-rich and shield-poor, so you need to irrigate the Floodplains to produce lots of excess food, and then use that excess food to make up the food-deficit for the citizens working on nearby food-poor Deserts/Plains/Hills -- which should all be mined for extra shields.
when I finally manage to built my first Settler, it is already 20 turns in game, AIs have already four or five cities each
At Demigod level and beyond, where all AI-Civs get 2 Settlers to start with, and a (>)30% discount on growth and builds, this might well be true. But at Warlord level? I find that
very hard to believe.
A Warlord-level AI doesn't get any extra units to send out exploring at the start, and requires twice as many shields to build (a new unit), and twice as much food to grow (a new citizen), as the human player does. So either "20 turns" is a massive underestimation of the point in your game when the AI-Civs have 4-5 towns on the map (200 turns might be more like it!), or you are leaving too many "goody-huts" (which look different from the Barbarian camps) for the AI-Civs to find*, or you are not playing at Warlord level (or I have misunderstood what you meant!).
*As
@vorlon_mi points out, you should try to send out 2-3 Warriors as early as possible, to explore the map in all directions. That will also mean that you will be able to find many of these goody-huts before the AI-Civs do, depriving them of the chance to 'win' extra Settlers or towns.