Probably to save material on the manufacture of the free plastic shopping bags in the US, these bags are small and thin compared to the ones in Estonia (which used to be free, but are now priced at significantly above production cost due to EU regulations on disposable plastic products). The equilibrium response of cashiers and customers to thin flimsy bags is to double-bag groceries, a practice unheard of in Estonia. After all, if one bag is sturdy enough, almost nobody will use two inside each other. I have successfully carried 10 kg in an Estonian plastic bag.
The equilibrium response to small bags is to distribute the groceries among many bags, especially the heavy or bulky items, for example to put each milk canister or large salad sack into a separate bag. Both double-bagging and the one-item-per-bag distribution lead to more bags being used in response to manufacturing each bag out of less material. It is an empirical question whether thinner, smaller bags result in less or more plastic waste overall. To incentivise reducing the one-time use of plastic bags and to encourage reuse, customers should have to pay for these, like in the EU.
One form of plastic bag reuse is as garbage bags (although it is only a one-time reuse, it is better than nothing). However, the flimsy free bags in the US come in bulk packs with the bottoms of the bags stuck together, so separating one from the stack often results in holes in its bottom an inch wide or more. The holes discourage many forms of reuse, including as trash bags, because small items (dust, crumbs, scraps) fall out. Partly the holes are due to the flimsiness of the material, partly to the way the bags are glued together to make a bulk pack.