Just to give a quick background on relays so you understand how they work. This is a standard 5-pin automotive relay:
Notice in the picture pin 30 has a wire on it that directly connects to a "steel plate" which is connected to pin 87a at all times.
Now, pin 85 and 86 go to each and of a magnet. The magnet has no charge unless it get energized. To do this, you must apply 12v to one side and ground to the other. It does not matter which side gets what, you can even give both ground or both power, it will not hurt the relay. All that matters is that once 85 and 86 get a positive and negative signal, the magnet in between them is energized. This make that "steel plate" which used to be connected to pin 87a, now pull down and switch contacts to pin 87.
At this point, whatever is connected to 30 now directly passes to pin 87 instead of 87a. As soon as you cut wither power or ground from 86/85, the magnet discharges and 30 switched back to 87a.
Now that you understand the internals of a relay, you can see that in your setup, pin 30 is the "source" which is going to power everything you connect to pin 87. This is why your 16g wire from the battery is what is directly powering your lights, regardless of what size wire goes from relay to lights. The power is not "amplified" inside the relay.
**NOTE that many people reverse 87 and 30. Meaning that I said 30 is your input from power source and use 87 to go out to your accessories... you can choose to make 87 your input and 30 your output, it does not matter. The are both connected the same no matter which way you look at it.
The 16g wire from your switch is in an independent circuit with the ground (pins 85 and 86) on the relay. Therefore, because a relay take a minimal amount of power to be "relayed" (switched, triggered, etc) then your 85 and 86 wires can be very small, as low as 18g or less even. However, the rest of the wires are more important!
With a full knowledge and understanding of a relay, you can accomplish an endless amount of electrical mods.
The first picture shows a relay at rest with the coil (85/86) not energized and the second shows what happens after it is triggered:
Here are a few of the various applications that can be done with relays and a little wire:
Headlights and Parking lights on with wipers (a law in some states). Notice is uses the wiper 12v+ wire to act as a trigger for the relay which then turns on the lights:
This is a weak positive input to a strong positive output (basically what you did in your system)
Convert a (-) signal to a (+) signal:
and the opposite...
The list goes on and on and on!
Notice that in almost all diagrams, pin 87a is rarely used. The purpose of this pin is to have a "pass-through" for certain setups. An example is used it for is in my headlights. I have a lighting system that requires me to cut power to my headlights automatically even without turning them off in the cab, and then send their power to a different set of aftermarket lights.
For this, the Low beam headlight (+) wire is cut in half. The car side is connected to pin 30 since it is my common input, then the light side is connected to 87a. This way, the lights function as normal without any further modification. Now, pin 85 is grounded, and 86 goes to the power switch for my aftermarket lighting setup. Once i turn on these aftermarket lights with their switch, it powers on the relay, disconnects my headlight wire which in turn shuts off my headlights, and uses their power to send out to my system through pin 87.
Hope you followed this. If not, oh well. Atleast your lights should get fixed after swapping out that one wire. Good luck.