You don't necessarily need 20mA, you just need enough to make however much light you want. 20mA would be the maximum, for an induvial, typical good LED, that comes with a datasheet with lifetime and current specifications.
10 ohms and 4.8V in is 20mA per LED (simple ohm's law problem). Except putting diodes in parallel like that and expecting them to split the current evenly rarely works out in reality, especially if they're not thermally connected, since forward voltage drops with heat and every diode is a little different, one ends up in a thermal runaway condition, taking the bulk of the current available, and then it blows and next lowest Vf diode starts taking the bulk of the load, until there are none left, as you've witnessed. Works alright for a throwaway flashlight that's expected to run maybe 100 hours at most though.
So more realistically, with this crude circuit, you want more like 100 ohms (20mA total), or perhaps even lower current/bigger resistor, you'll have to experiment as these are unspecified LEDs. 20mA is just a rule of thumb and the mystery parts in question may actually handle considerably less if you need them to go for years.
A better way to your goal might be to get some of the "1 watt" LEDs offered on ebay/ali pre mounted on a small circular PCB, and size a resistor for that, keeping in mind they probably need more than the included bit of PCB for a heatsink to run them anywhere near 1W, which is likely exaggerated (usually pretty safe to assume Chinese amps/watts are half size), but you could probably drive one at least as bright as those flashlights without issue, and you get some choice in color temperature as a bonus.