Just new tyre tread gripping and "jumping" (it means your old tyres were either slippier or the weather was!).
Nothing to do with RWD (in fact it would happen on a FWD and AWD with the same geometry and width of tyres etc).
Therefore upgrading to a C4 ain't going to help.
This "scrabbling" is normal on lock due to the greatest difference in turning radii between inner and out wheels and the camber induced.
Exacerbated by wide tyres and cold tyres.
If we are talking laws of physics, simply familiarise yourself with Ackerman Angles:
The steering system does implement an Ackerman compensation, but limitations due to complexity and robustness and catering for straight line feel and linearity etc, this is not perfect.
Even this little Lego version implements an Ackerman compensation geometry:
http://www.sheepo.es/2011/05/porsche-911-997-turbo-cabriolet-pdk_04.html
The above shows the idealised (zero contact pacth width) Ackerman compensation linkage.
There are many mechanical reasons why this cannot be implemented in reality.
The wide tyres is the most fundamental (as the inner and outer shoulder require different Ackerman compensation angles).
The jumping is when this differential force overcomes the tyres differential grip (across tread - greater force the greater the width or turning angle) - which occurs more likely under cold (hard) tyre conditions.