Regarding the $20/week thing ... it is doable if you live in a city or larger town with decent grocery stores and other food places so you can shop around. When I was in university recently, living in Edmonton, I was able to live on $20/week, and before that my ex-husband and I survived on $30/week in Winnipeg.
So in certain areas you can sustain life on quite low sums of money. However, I wouldn't want to live like that for any length of time ... I like my cheese and fresh fruit too much. And really ... why not spend more!! Why not enjoy a variety of food.
In other parts of the world, however, living on $20/week is quite difficult to do. I sense an assumption in this thread that if it is possible to live on $20/week in one part of the world, it's possible everywhere ... and that's not true. I've made comments already in this thread that the cost of food goes up once you get out into small towns or touristy towns ... and if you live in small towns or touristy towns, you'll also be paying more for groceries each week.
As it happens, Rowan and I live and work in a small town near a touristy area in Australia ... and it would be very challenging for the two of us to survive on $40/week. If we lived in Melbourne, that might be possible ... but not out here.
And this relates back to touring. If you are one of the lucky ones to live in an area with inexpensive groceries ... don't assume that you will be paying the same prices for groceries everywhere you go. Prices vary widely depending on several different factors.
There is even quite a difference in prices between the grocery store in the town where we live, and the grocery store in the town on the lake (right in the middle of the touristy area) a mere 25 km away. We've cycled to the town on the lake, and choke on the prices there if we want to buy a snack before heading back. I went to buy a 600 ml bottle of coke and a banana in one of the shops, and the checkout lady rang it up at $6. Yes ... $6 for a small bottle of soft drink and a banana. I decided I wasn't that hungry or thirsty and left it behind, but if you're on a tour, and it's late, and you don't want to cycle 1-2 hours to the next town to get something less expensive, you may be stuck paying those prices.
Regarding carrying food ... I have a guideline regarding the weight of the things I bring on a tour. The bicycle + all-the-gear-I've-got-with-me should not exceed 1/2 my body weight. I've done quite a bit of experimenting on tours, and when the bicycle + all-the-gear-I've-got-with-me exceeds 1/2 my body weight I struggle ... I struggle to maintain any kind of pace on flat ground, I really struggle on climbs, I struggle if I have to haul the gear between trains or up and down stairs in hostels. However if I keep the bicycle + all-the-gear-I've-got-with-me weight at or below 1/2 my body weight, I'm reasonably comfortable on a tour.
Because I don't weigh a huge amount, I need to be rather choosy about what I bring with me ... and hauling a week's worth of oatmeal, rice, flour, etc. isn't going to happen. I'd rather carry an extra pair of warm wooly socks!
Aside from which, I am the type of cycletourist who likes to try out the food in the area where I'm travelling rather than relying on the old, traditional basics.
And this is something the OP is going to have to consider. It has been asked before, but not answered:
What is the goal of this trip? Is it a personal accomplishment to cover a certain distance in a relatively short period of time? Or is it actually to see a bit of the country you're crossing?
If it is the latter, the OP might want to factor in the cost of stopping to see certain sights along the way, trying out a variety of food, etc.
On my 90 day tour of Australia, one of the best parts was the cruise out to the Great Barrier Reef. It cost quite a bit of money, and was time off the bicycle, but it was well worth it!