Weird strategy of going for 2-pair or higher

Are you talking about Holdem? or a different FL game?

Also your example is not really clear to me...

Broadway is a hand, namely two facecards. Normaly the term broadway is used to describe a hand pre-flop. Post flop he has High-card, Overcards, 1pair , 2pair etc...
If you mean Hero has TT and the boards is AKQJx with 3 clubs (I guess that is what you mean?). Then it all depends on how the hand developed if Hero has to bet.

example 1:

Hero is BU, V1 is UTG, V2 is MP V3 is BB:
V1 limps, V2 calls, Hero raises, V3 calls, V1,V2 call

Flop: A:c K:c Q:h
V1,V2,V3 check, Hero checks

Turn: J:c
V3 checks, V1 checks, V2 bets, Hero calls, V1,V3 fold

River: 2:s
V2 checks, Hero ?? ==> Hero should bet

However if the hand went like this:

Villains have similar positions:
V1: opens, V2 calls, Hero 3bets, V3,V1,V2 call

Flop: A:c K:c Q:h
V3,V1,V2, check, Hero bets, V3 calls, V1 calls, V2 3bets, Hero,V3,V1 call

Turn: J:c
V3 checks, V1 checks, V2 bets, Hero raises, V3 folds, V1 folds, V2 3bets, Hero caps, V2 calls.

River: 2:h
V2 bets, Hero ?? ==> here hero should call.

Since I have no clue of position, reads, betting patterns (Im not even sure if this the kind of hand you meant) I cannot give you a better answer

Sailer85, thanks so much for the detailed answer!

By broadway I meant the ace-high straight, AKQJT - you understood me correctly.

The first scenario you outlined is about how it happened. Thanks! I did bet in that situation, and you suggest it was the right thing to do - I think so too, although the villain seemed to very likely hit his flush draw (based on both betting patterns and physical tells), but I guess I should have bet still. He raised me on the river after I bet, and I called - should I have folded maybe? I hate folding on the river...

Again, your time is much appreciated!

Cheers,
Andrew

If you should bet/call or bet/fold kid a depends on the villain (or stakes if you play online). Seeing how the hand was played bij villain (ot that good), you should probably have folded on his raise.
However if he has shown you in the past that he can bluff in a spot like this then you should call.

If his body language and tells in live play show you that he is slowplaying with a flush then you can easily just check-behind... You have position so use the extra information that you have.
However be aware that you keep a unbiased mind. In my own live game a few of my friends will automatically fold on a flush-scary river "because he always has the flush", while in reality they only see it when he has it because he shows them the hand when he has it (and they remember the hands they see, not the ones he mucks).

I would just check call this hand if you think you opponent has a flush.

How good does your "suspicion" have to be? do you always check behind?

If villain had 2pairs or straight he could bet turn and check at river
Of course he could make the same with straight or flash
Of we suppose that from 10 times villain has 4 two pairs 5 flash and 1 straight if you check at river the last result = O
If you bet and 5 times he raises and you call and 5 times he calls you will loose 5*2=10 units and win 4 units
Total -6 units
So it is better to check