Given the combined operating ratio of most motor insurers is over 100%, it is wrong to think they make vast profits and are profiteering off the backs of motorists.
They are not.
There are a number of things that need to happen;
1 - spurious personal injury claims are properly investigated, and only those people that genuinely have an injury are compensated. The legal and medical profession are willing contributors to this because somebody with 'whiplash' will appoint an ambulance chaser who in turn will appoint a doctor who in turn will describe 'soft tissue damage' - this will invariably result in a 'go-away' payment of a couple of grand + another couple of grand in fees to the lawyer because it is cheaper than fighting.
I believe it is law in Germany that where accidents happen at 20mph or less, you cannot claim for whiplash - this would be a good start, on the basis that you can't get whiplash at such a low speed.
2 - Claims management companies should not be allowed to supply a hire car that is better than the car which is damaged. My brother in law's 2005 2000cc diesel Audi was involved in an accident a couple of years ago and he was provided with a top of the range X5 for a couple of months - the insurance company of the person that hit him had to pick up the tab.
Everybody has a common law duty to mitigate their losses.
3 - make the punishment for uninsured driving a proper deterrent. A long ban, a fine of 000's and seizure of the car.
I can probably think of more things - but these'll do to start.