The server uses a kilowatt of power or more (most of it in the CPU). But if the server is serving 1000 active users concurrently, and only 5% of the time you spend online is spent fetching ads, 20000 people staring at their screens get their ads from let's say 2 kW of server power usage, plus another 2 kW for all the equipment to get the data there... for a total of 0.4 watts per user.

These are completely eyeballed numbers, and could easily be off by an order of magnitude.

But your on premise gear (screen, computer, router) are likely by far the biggest factor.

One easy way to cross-check power usage claims is cost. It will only catch the most egregious bullshit, but it's easy. A random page I found claims that "According to the American Council for an Energy-Efficient Economy it takes 5.12 kWh of electricity per gigabyte of transferred data."

A Steam game with 50 GB would thus consume 256 kWh. Even if your 300 watt idle gaming rig, 50 Watt Router and 150 watt screen to watch the progress bar spends 2 hours downloading that, that's 1 kWh. Even at 8 cents per kWh, that means just downloading the game would cost someone (not you) over $20. Do you think steam would let you delete and redownload that game that you bought on sale for $10 as much as you want if between them and your ISP someone had to pay for $20 just in electricity, each time? Not the game rights, not the servers, not the connection, just power.