Haggling is traditional, expected and enjoyed in Turkey. It ranks as high on my list of holiday activities as sightseeing. If you go into the market and shops to look for souvenirs in Bodrum, you are going to have to play the game!
The most important thing is to decide how much you are prepared to pay for your (fake) watch or (fake) football shirt or (fake) designer perfume. Then let the shopkeeper quote first. Your offer should be much less than you are prepared to pay - as much as 25-50% of the asking price. This will probably be refused and then it's the shopkeeper's turn to bring the price down slightly - then you go in with another offer, slightly more than you first offered and so on - until you come to an agreement.
It could take quite a while and might involve a cup of apple tea ... you might need to walk away without buying anything but be gracious and part on friendly terms - the shopkeeper could begin to run after you and drag you back in and offer you a final price - if it's near enough what you expected to pay, don't let a lira or two get in the way of concluding the deal. Just remember what you were prepared to pay in the first place and more or less stick to it!
We bought a set of bowls for our villa from the market - having agreed a price for one, we then haggled for a set of four - the vendor was absolutely delighted to be doing business again with people with whom he'd already struck a deal a few seconds earlier! We all enjoyed the bartering and we were all happy with the outcome ...
Is this your business?