Imagine a free market with different consumers, who all have different limits of how much they are prepared to pay before they consume less or look for an alternative, and different producers, with different capacities and costs for production at their sites.
- If you assume an inelastic supply, then prices will rise until enough consumers stop buying. Basically, if there is demand for 100 barrels a day and production is 90 barrels, the price won't rise by 10%, it will rise until 10% of consumers give up buying.
- With an inelastic demand, if there is demand for 100 barrels and production is 90 barrels, prices will rise until producers find it worthwhile to produce 10 more barrels -- drilling deeper wells, extracting shale oil, etc. The new price will then be the price it costs to produce the 100th barrel, not the price to produce an average barrel. (Always plus taxes and profits, of course.)
In reality, both supply and demand are somewhat elastic, but neither can react immediately. Motorists who have purchased a gas-guzzling car cannot simply switch to a more efficient one, and wells and refineries have to be built before they can start.
So the best way to reduce gas prices in the US is to bring more suppliers with low production prices online, not increase the domestic output. Or to tell the Americans to drive smaller cars, but that seems unpopular.