Yes a Bay is much smaller than a Gulf.
A bay is smaller than a gulf and a sea is larger than a gulf
It sounds like you referring to a Canal.
A bay is a shallow, often in an arc, indent in the coastline; bigger than a cove but smaller than a gulf. A gulf is much larger and digs into the coastline further than a bay forming an arm of the sea. As by example of the Gulf of Mexico, the Persian Gulf.
GULF EXAMPLE: gulf of Mexico
both are of same size
The Hudson Bay is one of them.
Bay
A bay
The indentation of a shoreline that is larger than a cove but smaller than a gulf is called a bay. The famous explorer that was rescued in a bay was Ernest Shackleton.
== == A Bay
A bay.
A gulf. The word is derived from the Old French word "golfe" and the Greek word "kolpos" meaning "bosom, gulf."