Drag grows faster than speed, which is why cycling gets hard
The square law, and where the effort actually goes.
Cycling gets disproportionately harder the faster a rider goes because the air resistance pushing back grows with the square of speed, so doubling how fast a cyclist rides roughly quadruples the force fighting them.
Two effects multiplied together
Every extra bit of speed does two things at once. A faster rider meets more air each second, which on its own would raise the resisting force in step with speed. Each bit of that air is also struck harder, knocked out of the way with more momentum, and that effect grows in step with speed too. Two quantities that each grow with speed, multiplied together, give a force that grows with the square of speed.
Rolling resistance from the tyres behaves quite differently, rising only mildly with speed and staying small across the whole range a cyclist rides at. At a gentle pace the air barely registers and most of the effort goes into rolling and keeping the chain and wheels turning. At a fast, competitive pace the air has become the largest thing the rider is fighting, by a wide margin over everything else combined.
Twice the speed, eight times the power
On a flat road with no wind, riding twice as fast means pushing against roughly four times the air resistance. Power is force multiplied by speed, and that fourfold force now has to be delivered at double the speed, so the rider's legs must produce about eight times the power to hold the faster pace.
Power rising with the cube of speed also works in the rider's favour near the top of their range. Easing off by a tenth of their speed cuts the power needed by more than a quarter, which is why a small drop from a flat-out pace brings such quick relief.
Walking and running into a headwind
Walking at a gentle pace into a stiff wind produces a steady, manageable push against the chest. Breaking into a run into that same wind, at only twice the walking speed, turns the mild discomfort into a tiring effort within moments. The wind itself has not changed. All of the extra push comes from the runner meeting the air faster, the same square law a cyclist meets in still air. Stepping behind a wall or a parked vehicle makes the point from the other side, the push dropping away almost entirely once the body is out of the airflow.
Where racing cyclists spend their effort
Because air dominates at speed, almost everything a competitive cyclist does to go faster targets drag, with the bicycle's weight and rolling friction mattering far less. A low, tucked position shrinks the rider's frontal size, cutting the air struck each second. Riding closely behind another rider in a tight line, called drafting, lets the one at the back shelter in the already-moving air left behind the leader, sparing a meaningful share of the effort the leader spends pushing air aside.