Suspension of disbelief combined with artistic license.
The apparent in-story explanation is that the bridge is painted such that, from a "magic point" - the view of someone standing at the lion's head door - it blends in perfectly with the wall on the far side of the chasm.
To show us this intended effect, and make visually it interesting, a different "magic point" was used. Instead of bolting a camera to the fedora we are shown a view from above so we can see Indy standing there. That's the artistic license part.
We are given three shots of the bridge:
* An approximate POV shot - Indy look across the chasm. There is no physical bridge there.
* The overhead/camera move shot, and Indy crossing the bridge. The physical bridge is there.
* Indy tossing rocks/dust back over the bridge. This appears to have been done with a blue- or green- screen effect, i.e. the physical bridge was covered with blue or green fabric, the rocks were tossed onto it, and then the bridge is edited out c/o compositing, thus appearing "transparent". (IMHO, the effect looks jarring.)
Now the suspension of disbelief comes in because obviously, in-story, the effect would only work from the "magic point" - if you're too short, too tall, or move your head to one side... or if the lighting isn't perfect, the effect would be lost and the bridge would be perfectly visible.
That said, they did an amazing job with the set. In the overhead shot just before the camera moves, you really can't see it, and the wall texture is rough/striated enough that there's wiggle room in the "magic point".