Curry leaves and bay leaves are both whole leaves used in cooking, and American recipes sometimes suggest one for the other. They are unrelated plants used by opposite methods.
They come from different trees entirely
Curry leaf is the foliage of a small tree in the citrus family, and its aroma is closer to citrus peel and resin than to anything in a spice rack.
Bay comes from the laurel family and carries a resinous, faintly medicinal aroma dominated by different compounds. The two share only their physical form.
Neither has anything to do with curry powder, despite the name. Curry leaf is an ingredient in its own right, not a component of a blend.
One is used fresh, the other dried
Curry leaves lose most of their character on drying, which is why fresh ones are sought out and dried ones are widely considered a poor substitute.
Bay is normally used dried, and the drying is not a compromise. The characteristic flavor develops as the leaf dries, and fresh bay is sharper and less rounded.
The extraction methods are opposite
Curry leaves are dropped into hot fat at the start of cooking, where their oils release within seconds and the leaf crisps. The transfer is fast and the leaves are eaten.
Bay releases slowly into a simmering liquid over a long period, and the leaf itself stays leathery and is removed before serving. It requires time and water rather than heat and fat.
Using either by the other's method fails: bay in hot oil scorches without giving much up, and curry leaves simmered for an hour lose their aroma entirely.
Curry leaf carries a specific regional signature
The leaf is central to southern and western Indian cooking and largely absent from much northern cooking, so its presence marks a dish's origin fairly reliably.
Its combination with mustard seed and asafoetida in hot fat is one of the most recognizable aromatic signatures in Indian food, and no substitution reproduces it.
Getting them in the US
Fresh curry leaves are sold in Indian grocery stores, usually on the stem and in quantities larger than a single recipe needs. They keep in the freezer with better retention than drying gives.
Where they are genuinely unavailable, the honest approach is to make the dish without them rather than substitute. Bay, lime leaf and basil each change the dish into something else.