die Kurve
curve / bend
A bend or curve in a path, road, or track.
Der Autofahrer nahm die scharfe Kurve zu schnell und kam von der Straße ab.
The driver took the sharp curve too fast and went off the road.

curve
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curve / bend
A bend or curve in a path, road, or track.
Der Autofahrer nahm die scharfe Kurve zu schnell und kam von der Straße ab.
The driver took the sharp curve too fast and went off the road.

Pay attention to the prepositions and the case. Movement into a bend is expressed with 'in die Kurve' (accusative), while a location is described with 'in der Kurve' (dative).
Das Motorrad fuhr schnell in die Kurve.
The motorcycle rode quickly into the curve.
In connection with roads and paths, 'Kurve' (curve) is often combined with certain adjectives and verbs. Typical phrases are 'eine scharfe Kurve' (a sharp curve), 'eine weite Kurve' (a wide curve), or 'die Kurve nehmen' (to take the curve).
Sei vorsichtig, die nächste Kurve ist sehr eng.
Be careful, the next curve is very tight.
When 'Kurve' (curve) means a line in a graph, you often find the word in the context of data, economics, or science. Signal words are 'Anstieg' (rise), 'Abfall' (decline), 'Statistik' (statistics), or 'Diagramm' (diagram/graph).
Die Kurve der Infektionszahlen flacht langsam ab.
The curve of infection numbers is slowly flattening.
The idiom 'die Kurve kriegen' (to pull through / to manage it) is a fixed unit. The image behind it is a vehicle that almost has an accident but makes the bend at the last moment. It means to just manage a difficult situation.
Er hatte große Probleme in der Schule, aber im letzten Jahr hat er die Kurve gekriegt.
He had major problems at school, but in the final year he pulled through.
The meaning 'fan section in a stadium' is easy to recognize from the sports context. Words like 'Stadion' (stadium), 'Fans' (fans), 'Gesänge' (chants), or 'Fußball' (football/soccer) clearly point to this use.
Die Fans in der Kurve feierten ihre Mannschaft mit lauten Gesängen.
The fans in the curve celebrated their team with loud chants.
line
'die Linie' and 'die Kurve' can both describe something drawn or a course. A 'Linie' is a long, narrow mark; it can be straight or show a course in a general way. A 'Kurve', on the other hand, emphasizes a bend or a curved shape. For roads, you say 'Kurve' when the road does not go straight ahead. In graphs, 'Kurve' is natural when a line shows values and changes. Learners may confuse the words because a curve can also look like a line. The important point is: 'Linie' is more neutral, while 'Kurve' clearly says that something is bent or shows a course in a graph.

Vor der scharfen Kurve fährt der Bus langsamer.
Before the sharp bend, the bus slows down.
Zieh bitte eine gerade Linie unter die Überschrift.
Please draw a straight line under the heading.