How do I know why people prefer our competitor’s web site?

First let me explain the context: We have a web site that helps people to find jobs (e.g. https://www.careerbuilder.com/ https://www.glassdoor.com/index.htm )

In our country, there are some web sites that have the same purpose but there is one prefered by people. Obviouly it isn’t ours.

We want to do a research about the perception of our site and our competitor in order to find out why people prefer the competitor.

I thought about making some kind of A/B moderate live testing: Recruit several users, half will test our site and the other half our competitor’s web site. The same tasks will be performed by both.

But I sill don’t know if this will work.

Could somebody recommend me a test that help us to obtain significant feedback? Or is there another way to find out why people prefer our competitor’s site?

Thank you very much.