Did I say the moving platform?
Jump up and slightly left to give yourself more of angle, fire at the guy diagonally up + right, continue pressing right on the way back down to return to the platform, jump to the platform on the right.
If you want an alternative you can wait for the moving platform to be on its way to the right (and shoot the powerup box while waiting so it'll drop when you jump over), then leap to the right platform and immediately jump to the moving one. You'll have to pay more immediate attention to the timing of the left turret but you can bypass the worst of the grenade tosser this way.
Is this what writing FAQs for platformers is like?
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