Need help with rules - clock and move order

I am quite new to chess tournaments. And got into the situation there I am not sure what to do.

So we are at endgame of classic chess there we had 75+20 minutes.

I have 29 minutes left and my opponent has 40 seconds. Computer analysis gives me advantage during whole game, ~+1..+2.
As my opponent reached his last 5 minutes he stop recording his moves.
I continue. We did ~10 moves in such moves. And then after my next move following happened:
- I made a move taking his bishop by my king, and he has two pawns comletely blocked and I have pawn+bishop.
- I put my king to board and start moving my hand to the side with piece taken.
- My opponent immediately makes a move.
- Now I am confused and not sure what should I do - press clock, do next move or what.
- I am asking 'what are you doing'? opponent says 'everything is fine - just make your move'
- I am saying - no, it doesn't seem for me that everything is fine, let's call an controller.
- We called controller, who actually put position for ajudication.

I was pretty sure that I did right things, but tried to read afterwards and it seems that I was not right - such moves are allowed.

But I am very confused in such case. So questions are:
- Is that my opponent did actually allowed?
- If it is - than that was me who violated the rules, so technically game is lost?
- What's the point of clock then if opponent could make a moves while I am recording and pressing clock?
- What should I do in such situation in general - make a move and press a clock while he is moving piece/after he moved a piece?

My teammates all have different opinions on this so I am even not sure where to ask...

Thanks!

Your oponent has no right to touch the board nor any piece on it unless you pressed the clock (until you press the clock it's still your turn), otherwise he's moving in YOUR TURN which is against the rules. Plus with 20 sec increment the 5 minute rule thingy doesn't apply.
How did the arbiter rule things out?

He's supposed to wait until you press the clock to make his move, as you could use the time between completing the move and pressing the clock to offer a draw, make a claim etc. Not sure what the penalty is exactly, perhaps a warning or a 2 minute addition to your time. There's no reason that the game should be adjudicated.

Guys, that’s simply wrong. The following is possible and perfectly legal.

-Player A makes his move
-Player B makes his move
-Player A presses the clock
-Player B presses the clock

If a player makes his move his opponent has the move - according to FIDE, the clock plays no role for that. But: you can’t prevent your opponent from completing the move, meaning pressing the clock.

Actually, this sort of „premoving“ is considered bad style.

You could wait 5 minutes and then press the clock to confuse your opponent. :D

So, if I'm fast enough, I could make my first five moves while my opponent's clock is still running?

No, once again: the opponent is entitled to press the clock to complete his move even after your move. And you can answer one move „a tempo“ but then you have write down the moves.

@Sarg0n
The rules are very explicit about what "completes" the move:

Article 6: The chessclock

6.2
During the game each player, having made his move on the chessboard, shall stop his own clock and start his opponent’s clock (that is to say, he shall press his clock). This “completes” the move. A move is also completed if:
1. the move ends the game
2. the player has made his next move, in case his previous move was not completed.

So if instead of allowing this to happen (I can beacause of 6.2.a.2) I want to call an arbiter after you move in my time, regardless of how "bad" this playing style is. Provided the arbiter knows to read human language and he's not biased (which sometimes happens), he'll have to penalize the guy who moves in my time.

There’s consensus about that case amongst arbiters because of:

Article 1: The nature and objectives of the game of chess
1.1
The game of chess is played between two opponents who move their pieces on a square board called a ‘chessboard’. The player with the light-coloured pieces (White) makes the first move, then the players move alternately, with the player with the dark-coloured pieces (Black) making the next move. A player is said to ‘have the move’ when his opponent’s move has been ‘made’.

You have the move when your opponent made his move, NOT when he completed the move by means of pressing the clock. And you have always the right to complete your move even after your opponent had moved prematurely.

Ok, then I violated rules and should actually ask opponent to excuse me.

Thanks!