It's a pretty common problem. If you put a child seat on a normal bike, it will be very top-heavy and awkward. There are various options, though.
1) Get a front seat. That only works for smaller kids, but they're much easier to handle. The ibert has a 38lb limit. The bike will still be top-heavy, but at least you won't have to turn around.
2) Get a better kick stand. There are kick-stands made for cargo bikes that have two legs instead of one. They balance much better. I wouldn't leave a wiggly kid on one while you wander off for a snack, but they're fine while you're getting onto/off the bike and arranging the kid.
3) Get a trailer. You won't be able to have a conversation with your kid (and on the bright side - you won't have to have a conversation with your kid). But your bike will no longer be top-heavy, and you can put them in without worrying about the bike falling over. Best when paired with a mirror or something so you can make sure you have enough clearance when going through narrow points. An added bonus is rain protection.
4) As someone else mentioned, a trail-a-bike.
5) Get a bakfiet (or similar; Cetma, Bullitt, Metrofiet, etc). These are dutch-style box bikes with the box in the front. The weight in the box is very low, so the bike is not at all top-heavy. If you like having your kid up front, this is a good way to go. It's expensive, though; these start at $3k. Rain covers are available, too.
6) Get a mid- or long-tail. The most popular of these is the Xtracycle, but there are plenty of other options. Some of these are top-heavy, but some aren't; for example, the Xtracycle Edgerunner has a 26in front wheel and a 20in back wheel. That means you're up higher, while your cargo in the back is over the smaller wheel; hence a lower center of gravity, so less top-heavy. These don't typically have rain protection (though I have seen covers that people have hacked together). They're typically also pretty expensive, but less so than bakfiets. They start at $1k for the less fancy models, and the Xtracycles start around $2k.