Simple power equation question
If the total power across the circuit is 4W, how can I lose 20W of power over the resistor


You cannot get 1A through a 20&ohm; resistor with only 4W, so your assumption that it will draw 1A is wrong. If the voltage across it is 4V then it will draw 4V/20&ohm; = 0.2A, and dissipate 4V * 0.2A = 0.8W.

If the total power absorbed by the circuit is 4W then either there are other components drawing current in parallel to make the total 1A, or the voltage is being boosted before being put across the resistor. If the resistor is the only component then the current must actually be 0.2A, not 1A.