Why does my flight route look curved on the map?

Because the shortest path on a sphere is a curve when drawn on a flat map — the route is straight, and the map is what is distorted.

The geometry. The shortest distance between two points on a globe follows a great circle: the circle whose centre is the centre of the Earth. Stretch a string between two cities on a globe and it lies along one. But every flat map has to distort the sphere somehow, and the common Mercator-style projection stretches everything east-west increasingly towards the poles. A great circle drawn on that projection appears to bow towards the nearer pole.

This is why a flight from western Europe to the American west coast tracks over Greenland and northern Canada, and a flight to east Asia crosses the Arctic — routes that look like enormous detours and are in fact the direct line. Greenland looks vastly larger than it is on the same projection, for the same underlying reason.

What else bends a route:

Jet streams. Fast, high-altitude winds, generally west-to-east. Airlines ride them eastbound and avoid them westbound, so the fuel-optimal track differs from the geometric one and changes daily — which is why eastbound flights are frequently shorter in time.

Airspace restrictions. Closed or sanctioned airspace can force substantial detours, and has reshaped several long-haul routes in recent years.

ETOPS rules, which limit how far a twin-engine aircraft may be from a suitable diversion airport, constraining oceanic and polar tracks.

Organised track systems. Over the North Atlantic, aircraft follow structured tracks repositioned daily to suit the winds.

Weather and turbulence avoidance, plus volcanic ash and military activity.

Terrain, which matters where an engine failure would leave insufficient altitude to clear mountains.

A quick check: look at the same route on a globe, or on a map centred on the pole, and the "detour" becomes obviously the straight line.