How can you prevent lettuce from becoming bitter?

I think the key here is that you've been buying oversized lettuces. Bitterness of lettuce plants is not something under your control. When the lettuce matures (on the plant), it eventually starts flowering (called "bolting") as a way to perpetuate its genes. Once this sets in, the leaves turn bitter due to the production of certain chemicals, and acts as a defense mechanism against insects.

The process of bolting is irreversible and if it has commenced in the plant, it will continue on in the cut lettuce head (perhaps at a diminished rate). Big lettuce heads are invariably more mature than the smaller ones, which is why you experience a higher occurrence of them turning bitter. You could try the other suggestions here, but I cannot say for/against any of them as I haven't heard of any of them being used to control bitterness.

My advice would be to buy small to medium sized lettuces. Young leaves are tender and less likely to turn bitter. One might be tempted to buy the biggest lettuce on the shelf so as to "maximize" the leaves/dollar, but in the end, you have to decide whether the bitterness is really worth it.

I expect that the ultimate answer to your question is to eat your lettuce faster and restock sooner. I fear getting more than a week of "freshness" out of your lettuce may be expecting too much. Since you are storing it in a water bath, you can get an extra couple of days out of raising the acidity of your bath a bit. Try adding a little bit of lemon juice to water (1 tsp).

Remember to never cut lettuce with a metal knife. It will oxidize the lettuce and possibly create that bitterness. They sell plastic lettuce knives, but I used to work in the produce dept of a grocery store and our favorite lettuce knife was to get a plastic cake cutter (cheap/free) from the bakery dept.

This is a partial answer dealing with only one type of lettuce: There is no way I know of or have heard of to remove the bitterness in iceberg lettuce. There is a way however to avoid buying a bitter head of lettuce, which I practice:

When shopping for lettuce, I take use my thumbnail to make a wide scratch at the cut stem and smell it (use anything to make the scratch - the idea is to expose fresh unexposed cells). Bitter lettuce is detectable by smell and can be rejected.

Notes:
I have rejected whole bins of iceberg lettuce in the USA as bitter and used alternative types.

In the USA, the lettuce is now mostly wrapped in plastic at the grocery, requiring the peeling back of the plastic to scratch the stem.