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Normal just needs more powerful, reliable moves that can be coupled with STAB.
Bug needs an offensive buff. Make not so many types resist it, like removing resistance of Bug moves to Fairy, Ghost, and Poison. Make Bug be SE against Fighting.
Grass will probably see more usage because of the spore-moves buff to the type. I really don't see how to change it.
Make Rock take neutral damage from Steel. Make it resist Electric.
Ice needs to hit Water neutrally and make Water not very effective against Ice. Give it resistance to Dragon-type moves.
Finally, make Poison hit Fighting super effectively.