QUESTION
I have contacts that are so close together I'm either bridging them, or melting the neighboring one. What am I missing? 
ANSWER
Flux. You're missing flux, and lots of experience. I regularly solder with a tip that easily covers 10 or more pins at a time, but by carefully controlling how much solder is on the tip and using lots of flux I can get perfect solder joints. When soldering a TQFP or TSSOP package here's what I do:
- If the pads on the PCB are not super flat, then use some solder braid and suck up some of the solder on the pads to make them flat.
- Put the part on the PCB and carefully align the part. Don't rush this step, because if you mess this one up then nothing else will work correctly.
- Put a drop or two of liquid flux on the part. I use water soluble liquid flux, but others are successful with other liquid fluxes.
- Clean the iron tip and put a little bit of solder on it.
- Carefully touch a corner pad (NOT the pin itself) and let the solder wick up and tack down that pin.
- Repeat 4 & 5 for 2 to 4 corner pins. Reapply flux as required.
- Put some solder on the tip and run the tip down one complete side of the chip. Ideally you won't have bridges, but if the last 2 or 3 pins are bridged then you did good. Ignore the bridges.
- Add more flux (you can never have too much flux).
- Repeat 7 and 8 for all sides of the chip.
- Add flux, clean the tip on the sponge, and touch the tip to the bridge. Solder should stick to the tip. Repeat until bridges are gone.
- In bad cases, careful use of a solder wick can help. But be careful because it is easy to bend the pins and then you're, um, hosed.
- Clean the flux off. Some flux can be conductive, like the water soluble flux, so make sure it's clean (and dry) before powering up.
I should mention that my soldering iron has a straight, but angled tip. It's often called a chisel tip. This tip makes it easy to do everything from 1206's, 0402's, TQFP's, TSSOP's, etc. It just doesn't work for BGA's, QFN's, and some of the weirder things.
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