Max Moves cannot be leveled up directly in Pokémon Sword and Shield
Max Move power is determined by the Dynamax Pokémon’s base move and stats
To make a Max Move stronger, raise the Pokémon’s level
To make a Max Move stronger, use a Pokémon with higher base stats
To make a Max Move stronger, Dynamax a Pokémon with a better move in that slot
To improve Max Moves in general, use Max Soup to Gigantamax eligible Pokémon
To improve Max Moves in general, train the Pokémon’s EVs and IVs
To improve Max Moves in general, use items that boost the relevant attacking stat
To improve Max Moves in general, use a Pokémon with a stronger move type for the desired Max Move effect