That sounds about the same as I use for most jobs - the standard 15W Antex - but I also have a 25W and a couple of regulated/adjustable irons in the armoury (including one from Lidl/Aldi I've not tried out yet).
All I can think of is that your iron simply isn't powerful enough to heat the larger thermal sink above lead-free melting temperature (which is higher than tin-lead solder, making technique even more critical to avoid overheating components or circuit boards). A bog-standard soldering iron is only regulated by the wattage of its heater element versus the heat loss by radiation, convection, and conduction, so the temperature of the tip drops when placed in contact with the workpiece, and won't recover fully because of the extra thermal load (radiation and convection from the workpiece added to that of the iron itself). If the extra load is enough, the temperature may never get there... and keeping the input of heat going for too long risks damaging the workpiece. Ever tried soldering a wire outside in a breeze?
For other readers:
The soldering iron bit (or tip) is also important: a narrow point designed for fine work has a high thermal contact resistance so restricts the flow of heat to the workpiece. My preferred bit for general use has a bevelled "chisel" tip, which can be used on the flat to provide a good contact (or with its point for finer work).
Flux is important, because (when heated) it takes away surface oxidation to ensure new solder can merge with the base metal or old solder. If the solder you are using does not have flux included within it ("Multicore"), it is necessary to add flux separately otherwise the soldered joint will be unreliable (unless the surfaces are very clean indeed). Soldering works by creating a very thin layer at the surface of the base metal where the solder is alloyed with the base metal. Any contamination will interfere with this, and it is best to clean surfaces as much as possible first, even when using flux. The base metal is usually copper, sometimes brass - but if it looks silvery there is probably copper underneath with a solder coating (or at least a solderable coating).
Assuming the above is the case, and assuming you don't have an alternative iron, there are a couple of things you might try:
1. Add tin-lead* solder to the existing joint, which will gradually reduce the melting point of the existing solder by contamination. Add some, remove it (you have a solder sucker?!), add some more, remove it... Each pass should remove a little of the lead-free solder until you can desolder the component.
2. Reduce heat loss by warming the area around the workpiece, eg with a hairdryer (and keep it running so the surrounding air is also warm).
* Lead solder is only legal in very specific situations - including rework of an existing assembly originally made with lead solder (which lead-free solder won't work on). However: the regulations are only really enforcible for products sold commercially; lead-free solder is much trickier for DIY, and one has to use soldering tools that have never been contaminated with lead solder; and I made sure I have a lifetime supply of tin-lead multicore before it became difficult to get!
In my view, taking lead out of solder to please some environmentalists worried about lead contamination of waste tips, instead of controlling the disposal of electronics assemblies, is like throwing the baby out with the bath water.