Isn’t it worse for the environment to use plastic bags and then throw them in a landfill?

Perhaps not worse, but single use plastics certainly have their own drawbacks.

You can opt for biodegradable bags, which are usually made from vegetable resins or maize flour. However, even biodegradable bags may not completely break down in a landfill, due to the lack of oxygen. They will, however, decompose in a dog-waste-only composting bin.

A great alternative to bags is newspaper. Newspaper naturally decomposes, and you're reusing something you would have thrown away (or recycled) anyway.