Simple Solution To Stop Floods

Due to the amount of rain Britain has seen recently, the river banks are overflowing again and towns are flooding the same as they did in the early summer and have done every year recently.
Thanks to Global warming or the shifting jet stream or whatever reason we can come up with, the fact is that Britain is seeing more rain than is usual and therefore more pictures of poor villagers with their living rooms full of water.
The response is to strengthen river banks with flood defences which comprises of making the banks higher and higher but that doesn't go down particularly well because they are not aesthetically pleasing, massive lumps of concrete along the rivers banks may work but they look ugly.
Why not, i have often wondered, instead of building up the river bank, why not just dredge the rivers to make them deeper?      
Not viable say the Environmental Agency because they would have to dredge a few meters down and 'It must be repeated after every extreme flood, as the river silts up again'.
I accept that to dredge every river would be time consuming and expensive but surely they could dredge the ones that flood time and time again because that has to be better than the poor people who get flooded every time it rains enough.
With the extreme weather and floods becoming more the norm, villages along the rivers will become deserted as people move away as the cost of insurance goes through the roof, there are already people with no house contents insurance as they cannot afford it who lose everything to the flood waters.
Then there is the cost of the mopping up operation which runs into tens of millions each time so making the rivers deeper so they hold more water seems an obvious solution to me but then that could explain why I'm not employed at the Environment Agency.