I think there are several options.
First, I think that for the size loads you're talking about and for obvious weight balance issues, a "handlebar bag" is something to look at. Usually they just velcro strap to the bars and hook down at the fork dropouts, so are quite stable. The quick release mount type are easy to remove and carry with you.
A "frame bag" is also a pretty good solution for stuffing stuff on the bike, but gives up some convenience to a handlebar bag. Typically these are triangle shaped and strap to the upper corner of the seat tube/top tube junction. Capacity is generally smaller than handlebar bags, but some manufacturers, like Vaude, have long bags that run the length of the top tube and give more capacity. Because they can't be wide to avoid leg interference, they need to be packed, whereas a handlebar bag can often take a shopping bag (plastic one) stuffed carelessly in.
There are also "bento" or "front frame bags" that perch atop the top tube and are secured to the stem to prevent tipping. Lighter capacity than the above, and not necessarily quick on/off, though such designs may exist.
Another option, depending on how much seat post extension you have, is an seat pack. Some, like models from Topeak, can accommodate quite a lot by running down the length of the post. Again, depends on what you can afford in terms of space there, and may be in the way of the kid's face depending on seat location. Also, because the weight goes rearward, less than ideal for a rack mount seat IMO.
Lastly, for light loads, a couple of spaced apart self-adhesive hooks on the back of the child seat could be used to suspend a plastic shopping bag, or you could drill a couple of holes and use carabiners or something to achieve the same. I've done it, it can work, but it's not ideal from a weight distribution perspective, and unless you come up with the perfect solution, the bag can sway a bit annoyingly.
Anyway, I'd suggest starting your search here to see styles and get tips on pros/cons of each type of commercial bag: Standard Handlebar Bags | Bike Bag Shop | Bike Bag
Oh, I forgot to mention that there are also front racks you can install that widen options, from standard top loading types that can accommodate a "trunk bag," to wide, "porteur racks" that are designed for big cargo loads. Also, the "front pannier rack" would allow you to mount panniers at the center of the wheel, but really you need two to maintain acceptable balance by distributing the load, and that's probably more than you need, it sounds like. Again, check out BikeBagShop's front rack selection for ideas. Also take the time to look at the Nitto M18, which is a small, clamp mount front rack that doesn't require any braze-ons.