Are you using flux? Have you cleaned the traces with alcohol first?
At this small size, you don't even need to put much solder on the iron. Just whatever residue is left on it after you wipe the solder off. That should be sufficient.
Personally I use a chisel tip for soldering, the pointy edge of the chisel tip can make for far better pinpoint accuracy than the regular "pen" shaped tip, and conducts heat better due to its mass. If you have a iron, you won't need to turn up the heat very high, I can solder comfortably at below 300C. My secondary soldering iron is a crappier chinese all-in-one (that "yeehaw" heatgun + iron set) and doesn't conduct heat as good and I have to turn up the temperature, and it ends up not making proper contact and burning stuff everywhere, it's not any good.
But, I think those traces are just too close together. Perhaps you can put some Kapton tape on one trace to prevent solder sticking to it. But at that distance, I don't think it will help either (the board itself would transfer the heat from underneath the tape).
Maybe try using conductive ink, but those have far larger tips, so you'd need to tape down the entire area around each trace you repair (and hope that the tape won't lift the ink off once you are on the other traces, the ink takes something like an hour to set).
This is not an easy repair job.
