How to increase prestige effectively?

Sounds about right. Ensuring all your attackers have a maximum cp of half the lowest cp defender maximises your prestige score.

It can be done in two rounds. Basics: Take your lowest CP, which can defeat highest number of defenders. First defenders CP is decisive: if the gym starts at e.g. 2400, this is the lowest CP compared with you highest CP. If it is twice lower, you get 500 prestige, which is max! Keep in mind that there is no bonus for defeating whole gym, so don't push it. If the lowest CP is low, you have to defeat it with your low CP pokemon, which is annoying, because you always have to choose manually 6 low CP trash pokemon :(

by "complete clearing of a " I think he means that he has to beat each 3 mons 3 or 4 times, so 9-12 fights.
I think this thread can help you: https://pokemongo.gamepress.gg/q-a/understanding-defending-gym-prestige

You can take that gym down with other account and then replace with your. Training the gym take twice of time you taking the gym down even it is only 2k prestign.

I stopped training gyms since the update cause it would take forever but yesterday I had dinner at a restaurant that had a gym within reach and I was able to raise it up to level 10 starting at 3 in an hour. I didn't heal or revive anything , I just used whatever the game chose for me each try and even managed to clear it a few times up to level 8. It is essentially a loss of time but it not that hard to do if you really want to do it and have enough mons in stock. I'm lvl 29 with an army of 200 pokes above cp 1000 issued as filler. So really, just get a hold of tons of disposable high level pokes and stand there for a while. You'll get there sooner than you think

My own experience:
- If the gym is dominated with too strong pokemons, choose your six attackers that theirs CP are only half of the lowest defender's CP. Defeat the lowest one, run away and you earn 500 prestige point. Do it 4 times and enjoy your seat on top of the gym :D,
- If the gym is dominated with weak pokemons (those have many weaknesses), choose your attackers that can clear all gym defenders with as low CP as possible, then you can finish pushing the gym up in two round.
- Choose a rural gym so you won't waste your time and effort as gyms can be taken down too easily.

Unfortunately, it is just about impossible to get the maximum 500 prestige with a single attacking pokemon. The strength difference between the two is just two big. I did the calculations for electric pokemon attacking a CP 2400 Gyarados. If you go at it with a CP 1200 attacker, the bite attacks of Gyarados will take you out within 6 to 9 seconds. Jolteon goes pretty fast (6.2 s), as his level needs to be very low for CP 1200. Electrode and Elektrabuzz last longest (8.2 s). On the other hand, even considering the energy gained from damage taken, the attacker needs more time to complete even one charge attack (which in most cases is not even sufficient).

Thus, you need more than one attacker per defender in order to get the 500 prestige. This is neither time- nor potion-efficient.

Your bets choices in this particular case would be Jolteon, Raichu, or Eletrabuzz, all with Thunder (as will only be able to fire one charge move in any case). All of these will have to have a CP of about 1650. In this combination, the attacker is marginally faster to deliver his charge move than Gyarados is in knocking you out using only his quick attacks. Which leaves you with a gain of about 350 only, but barely achievable with a single attacker.

Other combinations are probably similar. Thus, your attacker should have about 70% of the defenders CP for efficient prestiging; 50% is too low.

Better to just take down an enemy gym. Take a couple buddies with you and the team can take down any gym quickly.

I really don't know why people have that much trouble...you DO have to dodge, but what I do is ALWAYS get 500 prestige to minimize fights...this can be very hard if people leave trash pokemon like 80 cp abras, but for most common defenders, look at the lowest CP mon (usually 1600-2000+) and choose a counter...which is 1/2 or MORE than 1/2. Expect to use potions/revives, but that's ok since you need to be fast and the gyms now have made it so I now throw those away anyways.

I can get 2k against a 2.2k Executor 4 times in like 5 minutes. Use 1k Arcanine/Parasect, against Vaps, use solar beam Parasects...or ivysaur, etc...

You NEED to build out a good prestige team. Then, what I do is permanently rename 4 or 5 10-15CP mons with a number in front so they are on the top...saves lots of time to pick them since I only need to do 1 fight and 1 1k parasect can knock out a 2k Vap.
With no all clear bonus, just fight the lowest mon 4 times and you're done...sometimes, I put a ditto and fight it with chansey/wiggly since that's faster than a full eggs fight if I need to make more than 1 spot.

Training a gym depends on what pokemon are in there. I just trained a level 3 gym to level 4 this morning in 2 rounds. Got about 1450 prestige both times for a total of 2900.

The reason was the three pokemon in the gym were all 2450CP+. This makes it easier to do. I can then usually get close to the max 500 per battle. Just need to use all 6 lower CP pokemon.

If I see a level 3 gym that has some 1200-1500 scrubs in it I just pass. It isn't worth it to power up. It will just take way too long. I like seeing a monster gym with all high CP pokemon, much easier to power up. I pass on the scrub gyms.