seems to me that the move 4...d4 gives white some trouble.
Caro-Kann defense, 2 Knights Attack- why is it not more popular?

Why would you open the position with dxe? You just gave white the bishop pair. Even the thought of taking, giving up the center for nothing is pretty bad. I think you should understand what is going on before you look at precise moves from theory.

Hey, I like this, gonna give it a few tries. I guess top GMs think it doesn't fight enough? hmm, I dunno.
Instead of 3...Bg4 putting the pin on white's king's knight, I prefer
3...Nf6. This refocuses white's attention to a battle for the center of the board where black has a chain in place reinforcing his pawn at d5. White's best chance here is 4.e5, attacking black's knight. Black has two candidate responses to the attack on his knight: 4...Ne4 and 4...Nfd7.

Wow, this is an old thread. I'll take a look at the analysis later, but the reason the 2N isn't more popular is that 2.d4 is simpler to learn and gives White an advantage. The edge is small, but clear.
There are GMs who play the 2N occasionally, but 2.d4 will probably always be more popular.

Black will have much easier time against The Two Knights Attack in Caro-Kann. I was black in one club game against a strong 2100 uscf who love the Two Knights Attack against Caro-Kann, I outplay him in the opening and achieve a solid advantage and won. White needs something more aggressive against Caro-Kann, Fischer give up on the Two Knights Attack and could not achieve an advantege, that is why its not play in the top GMs. It is true in chess club and below master level you see this opening system played, it is a way to avoid main line.

Schandorff gives 3...Bg4 in his GM repertoire and black equalizes easily.
Of course, if we're assuming black is equal in all major CK lines as well, your best bet is to be well prepared and try to test your opponent's prep. Personally, I still think there are ideas to be explored in the Classical mainline.
If you just want an out-of-book game... then The 2KA is a good way to avoid getting hurt by black, but don't expect the bishop pair to be worth much. In practice the stats show black doing well when the opponents are evenly matched.

Black certainly has no "easy equality" in the two knights. True, the bishop pair is not worth much: Not a penny more than a bishop pair...
Won't give any analysis here, you can do your own homework. Just a little "patzer game"- as a warning:
Personally, I play as white the variation of the "patzer" who had Black in the above game, but the two knights is just fine, and has way less to learn than 3.e5.
Did Nigel Short seriously not see that mate-in-one?

For what it's worth, Cyrus Lakdawala, The Caro-Kann Move-by-Move, says (with heavy abbreviating so as not to infringe on copyright):

So at the rist of asking something silly, is there something wrong with 3...e4?
At the risk of giving a silly answer... what's wrong with 3...e4 is that it's illegal

Tartakower beat Bogo as Black with 3...d4, but I agree with PFren, white has a nice game, especially after 4.Ne2 c5 5.c3! (Not 4.c3) Keres beat Tartakower with 5.Ng3!? in 1954, but I'm not sure it was due to the opening.

And there is of course "The mother of all CK traps"
I think this is Alekhine's game, isn't it?

I play the Caro-kann quite a lot, after E4,C6,NC3,D5,NF3,DXE4,NXE4 i play Nf6, i think that is the main line and the move Stockfish recommends.

I play the Caro-kann quite a lot, after E4,C6,NC3,D5,NF3,DXE4,NXE4 i play Nf6, i think that is the main line and the move Stockfish recommends.
This is indeed one of Black's best attempts- usually strong players refrain from taking at f6, and play 5.Qe2.
In this line as black I'm not especially fond of 6...Qd5 and I've been trying to make 5...Nxe4 6. Qxe4 Qa5!? work which Riazantsev and Grandelius have played. I imagine black's a little worse but I'd love to know your thoughts on it.
I've been playing the CK for a few weeks now, and I've gotten used to the classical, the advance, and the exchange (which is what 95% of people play against it). However, the 2 Knights System has given me trouble the 3 or so times I've faced it.