I have twins in nappies (well actually one is now potty trained but only last month) and i manage to spend less than £100 a week on shopping, including nappies, wipes and all household stuff.
I do meal plan though and don't buy any pre made connivence foods, everything is made from scratch. We eat meat though i do try and limit it to three times a week and make veggie meals/pasta etc for the other meals.
I refuse to buy berries unless from the greengrocer (and only if cheap) as my two could easily get through 1 tub of blueberries and raspberries each followed by some strawberries and then i wonder why i spent £10 on breakfast lol. I do love my avocado though so do get through 2 or 3 of those a week, however i do one big shop at Morrisons online once a month and they have 4 ripen at home avocados for £1 so i normally buy a few packets and they will last a few weeks (since they take so bloody long to ripen).
Summer is cheaper for us as we grow a bit of veg (potatoes, tomatoes, courgettes, carrots, corn and some berries etc) and we have a big herb patch so don't need to buy those either.
Fish and prawns are also really expensive so i imagine that is where most of your money is going. I would save it for dinners and not lunch. Maybe have tinned tuna (much cheaper in B&M's BTW, i got two 4 packs for £5 last week so half the price) or eggs for lunch as a cheap (er) and easy option.
The one who is still in nappies has Tesco value pull ups (£1.50 for 20) so that costs less than £5 a week for nappies and wipes.
Meals here are normally, meat and veg, stir fries, fritatas (cheap and cheerful and veggie), pasta (meaty or veggie), we eat a lot of curries (again both meat and veggie versions) paella is a popular one, soups, stews and dumplings, chillis, slow cooker casseroles.
It is possible with planning and shopping around. I am aware that not everyone has the time to do so though.