A basic calculation shows that 60W at 12V will pull a current of 5 amps through your circuit - wires, switches, connectors etc. So any part of that which is rated at under 5A will not be adequate. Remember that fuses are present as a deliberate weak point in a circuit, and when the current is too high the fuse melts.
But electrons zapping along a circuit don't know this - no-one told them basic physics. As far as they're concerned, they'll heat up EVERY part of the circuit, and the part that gets hottest soonest will melt - whether that is the fuse or a contact in the switch or any other part, wires included. So make sure the fuse really is the little tubey thing you pop in the fusebox, and not the wiring itself.
If you don't have wire of adequate gauge and you're not sure about keeping the right colour codes and so on, an easy solution is to take a length of similar gauge wire insulated in white (so colour code isn't confused, so don't use black [earth] or red [live 12V]) and run it along the same path as the existing wire, taping the two together every few inches so they are kept together and the colour coding of the original wire is maintained for both. This will effectively double the capacity of the wire, that is, halve the current (and hence the heat generated) in each wire, all else equal.
When it comes to the switch, fitting a relay is the best practice, but switches are generally pretty robust and will carry a significantly higher current than the wiring attached to them. The 'weakest' part of mechanical switches is usually the contact points, and if you cannot fit a higher current rated switch, if for example you need to maintain physical fit and or aesthetics, it is often possible with older switches to dismantle them and ensure the internal parts are properly lubricated, free from corrosion and clear of old dried up grease and general dust and grot. If the contact points are raised dimples on a copper strip, you may be able to flatten these slightly with judicious hammer blows so as to give a greater contact area and hence higher current carrying capacity at the contacts.
There's nothing magic about a relay - it is simply a switch with massive great contact area, but using one avoids the need for fitting a huge switch and thick heavy wiring to the dash area where space for such things is very limited.
Apologies if I am telling you how to suck eggs, but hopefully this may help others
