Why does iterating in different ways produce different solutions?

You are aiming at fixed points of two different functions.

Some fixed points are attractors and some are repellers.

If you graph your functions you see as you iterate, the points tend to the attractor and stay away from the repeller.

For $f(x) = \frac {2^x}{4}$ the point $x\approx 0.3099069324$ is an attractor and $x=4$ is a repeller.

For the other function it is the other way around.

An "exact" solution may be obtained with use of the Lambert W-function which is defined by $W_{0}(x) e^{W_{0}(x)} = x$ for $x \geq -1$. This is seen by the following: \begin{align} 2^{x} &= 4x \\ 4 x &= e^{x \, \ln2} \\ (- x \, \ln2) \, e^{- x \, \ln2} &= - \frac{\ln2}{4} \\ - x \, \ln2 &= W_{0}\left( - \frac{\ln2}{4} \right) \\ x &= - \frac{1}{\ln2} \, W_{0}\left( - \frac{\ln2}{4} \right). \end{align} By calculation this gives a value of $x \approx .30990693238\cdots$.