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tl;dr/conclusion: just flush your urine. this post isn't accounting for the increased used of cleaning products caused by urine stains, so really all of these numbers are even lower than they appear. if you actually want to reduce carbon emissions and energy usage, your energy is better spent on political actions. and if you insist on reducing consumption, stop buying avocados and turn off your air conditioning - that's gonna have magnitudes more of an effect than not flushing.
Let's do the math. Average toilet uses about 1.6 gallons (0.00606 m³) of water per flush *ᵃ. Flushing doesn't actually use energy, but it takes energy to both
1) treat the water before it gets to your house and
2) to pump that water to your house
It takes about 1 kWh to treat 1,000 gallons of water (3.785 m³) *ᵇ. That translates to about 0.0016 kWh per 1.6 gallons (0.00606 m³). So it takes 0.0016 kWh in order to treat the amount of water your toilet flushes in a single flush.
It takes about 3,300 kWh to pump 1 million gallons of water (3785 m³) *ᶜ. That translates to about 0.00528 kWh per 1.6 gallons (0.00606 m³). So it takes 0.00528 kWh in order to pump that 1.6 gallons (0.00606 m³) of water per flush.
If we add those two together, we get 0.00688 kWh needed to pump and treat the water every time you flush. Let's try and put that amount of energy into reference.
So, let's see how much energy we would save if every single household in America flushed one time less per day. There are about 125.7 million households in the US *ᵈ. That would translate to about 864,816 kWh per day, assuming every household flushed exactly once less than they usually do. Let's put that figure into reference.
sources
a: https://www.savingwater.org/indoors/toilets/how-much-water-does-your-toilet-use/
b: https://www.esmap.org/sites/default/files/esmap-files/FINAL_EECI-WWU_TR001-12_Resized.pdf--
c: https://css.umich.edu/publications/factsheets/water/us-water-supply-and-distribution-factsheet
d: https://www.census.gov/quickfacts/fact/table/US/HSD410222
e: https://en.wikipedia.org/wiki/Electricity_sector_of_the_United_States#Electricity_consumption
f: https://digiconomist.net/bitcoin-energy-consumption
g: https://www.energy.gov/energysaver/air-conditioning
fairs man, you convinced me!