Amazon.com Inc. has trained people to buy everything from major appliances to daily staples online. Now it is having second thoughts about some of those sales because they don’t make money—and is pushing big brands to change how they use its site.

Inside Amazon, the items are known as CRaP, short for “Can’t Realize a Profit.” Think bottled beverages or snack foods. The products tend to be priced at $15 or less, are sold directly by Amazon, and are heavy or bulky and therefore costly to ship—characteristics that make for thin...