are these machines new or old?
have you had the forks/mast/carriage inspected for wear with the proper gauges?

Fork bending (flexiing) is also a functon of the carbon content in the steel, meaning the higher the carbon content the less flexing will result(assuming teh same fork section size). On the down side the higher carbon steel fork will have a tendency to break rather than bend when teh fork heels are worn or teh forks are subject to overloading.

Typically low carbon steel forks come countries in Asia and are of a larger fork section size than US forks of the same capacity. Many of the forks used in the US come from one supplier.

Joe, the bending of forks should not become an issue if the load is within the rating of the fork. The answer of increasing the cross section will increase the capacity of the forks. The relationship is 1/4 inch in thickness equals about the same as 1 inch in width. BUT...
It sounds to me that the real problem is you are using mass produced forks made from rolled flat bar. These forks, sold by almost all fork suppliers here in the US, are NOT heat treated fully and they are NOT fully forged. They are hardened only at the bend. Heat treating fully to the correct hardness spec over the entire fork will reduce the flex of the fork dramatically.
If you use a fully forged, fully heat treated fork, such as the ZYatWork brand sold by Fork-Co USA and Trading Point Products, my bet is you will not encounter the flex/springing problem using the same fork you currently use. That is, unless you are now using a fork that is very much too thin and under capacity as it is.
You won't find this type of fork from the sellers who deliver flat bar or plate plate forks. They don't hammer forge flat bars. ZYatWork hammer forges round bar alloys to make the forks...100% forged. You can reach them in the USA at 281-648-2723.

Thank you AndyPandy, as your "wider forks" solution seems to be the most likely way to solve this particular problem.

As you are referring to the natural flex of the fork, i would suggest fitting a heavier duty fork (larger cross section) might help. For example if you are using 100mm x 40MM (2500Kg at 600mm) at the moment you could change these for 100mm x 50mm (3500Kg at 600mm) which would reduce the flex you experience. Of course there are many other fork sizes available, just pick one to suit your application.

Thanks for your response, fixitandy.

By "bending forks" I intended the word "bend" to mean "flex" or "spring".

When picking up heavier loads, the flexibility engineered into the spring steel of the current forks causes the load to "lean" forward as it is picked up. Re-engineering the rack system is not feasible. Turret trucks are also not feasible at this time. It is possible to train operators of counterbalance rider trucks to handle the "leaning" loads without hitting the rack shelves, but because the new palletized load heights leave so little clearance, training for the required precision takes too long to accomplish. I thought there might be an alternative fork supplier that might have less "spring" in their spring steel forks, or some other solution to this load handling problem.

Your question is a bit unclear, when you say "bending forks" do you mean that the machine in question is a "bendi" or similar machine with a rotating mast? If that is the case, a drexel might do the job better than a bendi, but because a drexel mast tilts from the bottom similar to the bendi, it might not be a lot better. the next step up in the very narrow isle machines would be a crown or raymond turret or swing reach truck.