It’s not as unusual as you think... But Chris is correct, the Lead should be underneath the shingles in a step flashed install, from the bottom up. (each piece over lapping the lower piece)
I can’t really tell from the 1 picture, if you had a couple more it might help. The problem can be multiple things.
- If there are nails holding the lead down, they could be leaking.
- If the caulking at top is not sealed to brick and lead properly, that could be leaking.
- The back of the chimney lead is also on top of the shingles, which will allow water underneath and will cause a leak.
The recommended action is to remove the flashing and shingles around the chimney and do a rework with Aluminum Step and Cap/Cover Metal.
See attach picture for an example.
Notice how the metal at the low point of the chimney is the only exposed metal on the roof and there are no nails exposed.
Hope this Helps!