Avoid letting Bowser make more than three full rotations before throwing him; afterwards, he will spin too fast and become more difficult to aim.
In order to trigger the text after Bowser is defeated, Mario must be facing towards and be within a short distance of him.
The "whoosh" sound that plays while spinning Bowser plays when he faces north, and is not indicative of whether he is pointing towards a bomb. Try to focus more on aiming visually.
It is highly recommended that runners who play using the original Nintendo 64 controller switch to a different controller for Bowser fights, as rapidly spinning the N64 controller's stick will wear it down. Recommended controllers include the Hori Pad Mini, a Retro-Bit Tribute 64, or a GameCube controller via a Raphnet adapter.
Bowser in the Dark World
Moving Bowser
To avoid an air throw, let the stick go neutral right before pressing B to throw.
In this battle, Mario cannot grab Bowser until the platform stops teetering. Avoid groundpounding during this period, otherwise Mario may clip through and fall into the lava underneath.