Of course, you will be suffering whip anger and decreasing your happy cap. There are a few things you can do to work your way around that.
First thing is : whip early.
The earlier you whip, the earlier whip anger starts to fade. Then, you'll be able to afford the next whips earlier.
A situation you would want to avoid is to whip at happy cap, working all improved tiles and lose control over those tiles because of the whip anger.
Second thing is : whip several pop points at a time.
You get 1 unhappy per whip and you get 30 hammers per pop sacrificed.
In the early game, it can be fine to whip a single pop (2 to 1) because the city has a lot of happy margin and there is time for the anger to fade. A city at happy cap whipping 1 pop would simply lose control over tiles.
So, getting 1 unhappy (for 10 turns) is generally associated with getting 60 or 90 hammers worth. The hammers to unhappiness ratio is more favourable and the whip anger has more time to fade while the city regrows.
Third thing is : whip useless tiles.
You can whip unimproved tiles or food negative tiles like mines.
You can also whip unhappy citizens. A food rich city (more than +6F per turn) should not be afraid to grow into unhappiness. An unhappy citizen is -2F per turn (man's gotta eat) but it's also 30H of stored production for the whip.
Growing into unhappiness and then whipping is a way to minimize the number of tiles you lose control over.
A city doing +10F could easily grow into unhappiness, do +8F, grow into a second unhappy citizen and then instantly whip 3 pop (2 of which are unhappy) for 90 hammers. You would want to whip asap, in this situation, so as to avoid stacking too many -2F from unhappies.
Fourth thing is : stagnate at happy cap or higher.
Your city can hire specialists or build settlers/workers to avoid growing further.
Note that the unhappy citizens do not eat (not -2F) when the city is building settlers/workers.
So, if you're accumulated too much whip anger and are waiting for it to fade, now is a good time to contribute a worker/settler to your Empire.
[Note that a high food city (pigs+floodplains) is very well-suited to grow to happy cap asap. If your goal is fastest workers/settlers, however, a cow/copper city would be stronger. It's the tile yields that one should be looking at.]
Fifth thing is : use overflow to convert Food (building settlers/workers) into Hammers (building infrastructure).
As a general rule, the early settlers/workers should be produced asap if/when whipping : minimal overflow. If you want to whip a worker from size 2 to 1, you just need 30 hammers of production.
Now, it's different if you're preparing to build a Library.
You might not want to sacrifice 3 pop points to get the Library and lose control over precious tiles.
This is the time when you set up a settler/worker whip with max overflow, which will give you something like 40 hard hammers into your Library. Then, the building can be 2 pop whipped or completed with some over whips and overflow.
Example : A settler costs 100H. If you whip it just under 70H, 2 pop will be sacced. Production will raise by 60H to just under 130H. Then there is the production for the turn to something like 140H/100 and you have 40H of overflow to place into a building.
Sixth thing is : timing your whip.
Sometimes you want to whip asap. You're building a worker or a settler or you want to start your fading counter for whip anger.
Sometimes you have a 1-3 turns window to choose. And that will be related to your food count.
Example : You are planning a whip from size 4 to 2. The eventual food bin will be 24, yur food count is 16, your food surplus is +6. You can whip right now and reach 22F/24 or you can work your tiles for a turn longer, whip next turn and reach 28F/24 (regrow immediately).
Whipping loses control over tiles. Assuming infinite happiness and a whip cycle (max prod vs rush item), it is always ideal to whip and regrow instantly.
Now, if the city grows onto food negative (less than 2F) or into unhappiness, that can be taken into account, too. Whipping earlier will raise the food output of the city and allow it to regrow faster (hence my example on whipping 2 unhappy citizens asap).
So, solutions.
Where TaPele is very right is that those high food-low hammer cities need to be carefully managed and run poorly on auto-pilot.
Production has to be planned. You need to know where your hammers come from and how you intend to build each item (which is generally true for all cities and build orders).
Final note : If you've read carefully, you will have understood that starting with Animal Husbandry on this map, as I have done, and delaying Bronze Working for the longest time, is a highly dubious line and far from optimal. Being CHA gives a lot of happy margin for early whips. The CHA/EXP combo is particularly strong for 2 --> 1 Worker/Granary whips (or just plain chops).
It doesn't help that pigs and horses are the only 2 AH resources. It would be a little closer with several cows, which we don't have.
I think AH vs Bronze is an interesting question that the opening screenshot raises but : just being EXP makes it extremely valuable to acquire Bronze + Pottery (for said quick Granaries).
Hope that helps