Some manufactures are trying to yet build some naturally aspirated engines, but they are the last of their kind. Some examples: Lamborghini Aventador, Ferrari 458 Italia/Spider, Ferrari FF, Ferrari F159 (599 replacement), Porsche 911 Carrera /S (991), Porsche Boxster/Cayman /S (981), Aston Martin One-77, Lexus LF-A, Toyota FT-86.
But some others are are pioneers in the new era of turbocharging, as BMW M and McLaren. Now we can't say they resolved any of the issues, but they are getting closer.
The fist Ferrari turbocharged engine that we are to see are to appear on the up-coming Maserati Kubang and Maserati Quattroporte. From what I heard, the Maserati engine built by Ferrari will be a 4.0l V8 with about 500 PS. Now this engine will be tuned to fit a Maserati, so high-revs and instant throttle-response is not such an important thing as it would be in a Ferrari, just as it isn't as important in an M5 compared to an M3.
I do believe, there is a way to get read of turbolag, indeed there already is, and it is called anti-lag, it is used in rally cars, but this cannot be used on streetcars.
For me the greatest problem is the sound, and for this unfortunately there is no solution.
M3 CSL (E46), LF-A, 458 Italia and 599 GTO are some of the best sounding cars on the planet. Put a turbo, they'll be very far from that.