Don't fall into the trap of thinking that reinforced paper bags are the magic solution. Sure, they hold up better than thin plastic, but if you fill them with heavy stuff or they get wet, everything falls apart and you end up buying more. What really works is combining options depending on what you need in that moment.
For grocery shopping, the cheap cloth bags they sell by the kilo at discount stores or at the supermarket itself are your best bet. You don't need the fancy designer ones that cost an arm and a leg, those basic ones in simple colors last years if you don't beat them up. The initial investment is minimal compared to what you were spending on free bags before, and you recoup it quickly. I ended up buying like 5 or 6 a while back and I'm still using them, so in the end it's cheaper than constantly paying for new bags.
For one-off situations where it's not worth lugging a cloth bag around (a small purchase, something that might break), reinforced paper bags work fine but use them smart: don't stuff them to the point of bursting and keep heavy things at the bottom. Another option that a lot of people overlook is the boxes that the supermarket itself gives away or lets you use; ask the cashier, sometimes they have cardboard boxes from products that they're going to throw out anyway. I sometimes bring boxes from my house, seriously it works.