I would assume neighbours might be putting bags in your bin.
Buy bin locks.
Separate your food waste properly
Separate your recycled waste properly
Crush all bottles/cans/boxes to minimise space.
We are a family of 6. We've recently lost one of our bins as we didn't wants to pay the extra when they changed the compostable bin to a chargeable, garden only instead of food waste/compostable items such as shredded card and paper. Our garden wasn't big enough to justify the cost.
However we are fortunate enough that our council allow an extra, full size recycling bin for families of 5 or more.
If we don't sort and crush rubbish properly then we can overfill a bin in a fortnight (one week recycling waste, one week tip waste). Even then, we are able to squash the bins down a bit more with the judicious use of a broom so the lids close.
It's not impossible, but it does require a little bit of effort.
It seems like your problem is, aside from any potential bin dumpers, you don't want to make the effort when it's comes to doing your bit.