All the flight booking websites tell you they find the cheapest flights.
What they really mean is that they find you the cheapest airline that pay them a booking fee.
Often there are cheaper options, but its knowing where to look.
One site I regularly use is Google Flights.
The reason I prefer their site is they do tell you if there are airlines that fly the route, even if they don’t pay a commision.
These are likely to be the cheapest fares for the route.
For example check this screen shot from a flight search from Chicago to Los Angeles.
You will see that there are a couple of flights listed by South West as ‘Price Unavailable’ so you need to then go to their website and check there.
NB Southwest is a large American budget airline that doesn’t pay commission to websites. Unlike many budget airlines their standard fares booked on their own website include checked bags.
Disclaimer I haven’t been paid or received any other reward for this post