After you pass the two zingers, there is an auto-barrel that sends you into a barrel cannon that will spin. Try to aim down and it will shoot you into an Auto-blast barrel. That one sends you into a trail of bananas and into another blast barrel followed by one more trail before a final blast barrel shoots you upwards. You'll be right at the exit.
This could be where the game gets harder (at least this world’s last level - Rope Bridge Rumble, isn’t so hard) [though the next two worlds - Kremkroc Industries Incorporated and Chimp Caverns - are hard].
