Tilly,
The best place near the airport for furniture (if you have to buy several items) is kisikkoy. It's a large industrial furniture making village with show rooms. You can park up (for free) and walk around all the different shops.
To get there leave the airport and you should come to roundabout, you want to head towards AYDIN or KUSADASI and continue driving for approx 10 mins, this will bring you to a smaller roundabout and on the right hand side you'll see what looks like an industrial area, called kisikkoy, there's a large metal sign.
You can purchase everything here, right down to coffee tables and up to self designed bedroom furniture. We had our lounge sofa made to our own specs, width, length colours...you can choose everything your self if you don't see what you want ( a friend also had a drinks cabinet made in the style/colour/design she wanted)
KISIKKÖY MOBİLYA ŞEHRİ
Make sure you barter on the price, you can negotiate here. If this sounds too much just drive down to gazemir, there's several furniture shops along the very long main road.
Good luck,
Charlotte