I have a quantitative answer which is a thought experiment avoiding all but the simplest equations.
An object going from velocity v=0 to v=1 needs to be pushed or pulled in some way. In my explanation I will use the same method to push the object from v=0 to v=1 then from v=1 to v=2, then v=2 to v=3, etc. I will show how the energy of movement embodied in the object goes up from 0 to 1 to 4 to 9, etc.
Start with two identical balls, m1 and m2. Between the two balls is a spring, s1, which is held in compression. Assume the mass of the spring is very small. The potential energy in the spring is PE=2 and all 3 actors have velocity v=0.
A. v=0. All objects have 0 velocity so kinetic energy KE=0.
B. v=1. Release the spring and m1 shoots off to the left with velocity v=1. m2 goes in the opposite direction with v=-1. The kinetic energy of both balls is the same and is KE=1 because all of the potential energy of the spring has been transferred symmetrically to the balls.
C. v=2. Now place another identical ball, m3, just to the right of m1 and also travelling at v=1 and with a compressed spring, s2, between them. 512
Nothing has changed about m1, it is still happily travelling at v=1. So what's the total energy of the m1, s2 and m3 system? It's 1+2+1=4 being m1's KE, s2's PE and m3's KE.
Now release the spring and m1 shoots off to the left with v=2 and m3's velocity goes from v=1 to v=0 making its KE=0. Because we've said that the mass of the spring is very small so its KE is almost zero then all of the energy which was in the system before the spring was released is now in m1. So the KE of m1 is KE=4. Phew, KE is proportional to v squared!
D. v=3. Simply repeat the process to make m1 go from v=2 to v=3 by pushing off another identical ball, m4. First, work out the total energy of the two ball and spring system before the spring is released. It's 4+2+4=10. After the spring is released m4 has v=1 which we've established is equivalent to KE=1. So m1 has the remaining energy of the system which is KE=9.
E. v=4. Repeat the process. Energy of system before the spring is released, 9+2+9=20. KE of m1 after spring is released, KE=20-4=16.
I'm not happy with assuming away the mass of the spring so a neater explanation has a spring attached to each ball and the balls interact via their springs which are in contact. 621