Amazon.com Inc. this week filed lawsuits targeting sellers allegedly listing counterfeit goods on its website, publicly cracking down on an issue that has caused increasing friction.

The two lawsuits, Amazon’s first legal challenge against its third-party sellers for bogus products, pertain to fake versions of the Forearm Forklift, a strap system that allows users to lift heavy items, and TRX Suspension Trainer workout bands and products. The second lawsuit was filed jointly with Fitness Anywhere LLC, the company behind the...