Next to tires, cassettes and chains are the fastest-wearing items on bikes. As Dan333SP said, the problem might be occurring because you replaced the cassette but not the chain.

If you get a new chain, let's hope that the cassette hasn't been damaged by the old chain to the point where the cassette will have to be replaced again.

Also, many people who mention that they're very strong riders mostly use the smallest two sprockets on the cassette, and those are the fastest-wearing sprockets, since they have so few teeth. Once you get the problem corrected, you should consider using the big ring in front with sprockets other than the smallest in back for most of your flat-terrain riding. The fastest riders train by spinning moderate gears so that they can spin big gears in competition.