It is something I see continuously in American TV shows and films. Someone will come back from the shops (or grocery store) and everything will be packed into the most flimsy looking brown paper bags with additionally weak handles.

This is ideological, most places do not carry paper bags, the depiction in commercials/films harkens back to "the good ol days" when we didn't already fuck up the earth.

How do they not split? What happens when you put heavy cans into them? It also appears that they often have no handles at all and you have to just clutch them against your chest.

Paper bags are much stronger than plastic, without handles though we simply carry them in our cart back to the car. For heavier items, those are double bagged. Places like Trader Joe's will encase the paper bag with a plastic one for you to carry easily.

Do you not do plastic bags? It just seems like a disaster waiting to happen! Thanks in advance!

Oh yes, Americans do plenty of plastic. Some of the more progressive states use reusable bags regularly though. For example when I lived in Northern California I had 8 canvas bags and about a dozen mesh bags for produce. There is a lot of debate about using those but I am of the mindset that a dozen less plastic/paper bags floating around per trip is a good thing.