The first one is the Asian currency crisis, and the second is the tech crash.  Prices were not particularly high in either case (according to BP on an annual basis, WTI was $20 in 1997 and $14 in 1998.  In 2000 it reached $30).  It seems clear to me that both events are demand-side related.
Weird, we both look at it and come to 2 completly different conclusions at the same time!

One says supply and one says demand.

I'm no expert so I must assume you are correct.

Well, that's handy.  Calculated Risk  (posting at Angry Bear) has a nice summary of Fed transcripts from May 2000, right as things were about to go South.  Here's US GDP changes:

You can see that GDP slows down really a lot in late 2000, whereas the oil production in OilCEO's graph above doesn't slow down until the end of 2000.  So GDP changes preceded oil production changes.  However, in the current case, oil production growth started to slow down in late 2004, and GDP growth was good until Q4 2005.

Hence it appears that the causality is reversed in the two cases.

We'd have to look at Asian data to see the 1998 events, but I don't have it to hand (nor time to hunt now).

Stuart, I'v been futzing around with price all night trying to somehow add it to this graph. Unsatified at my attempts to somehow relate the two in a chart that shows promise for a correlation. Any thoughts?
In Excel, under the Chart Menu, choose "Chart Type".  Select the "Custom Types" tab, and then choose "Lines on 2 Axes".  

Also, if you've already got a chart you can add another series on a second axis somewhere under the Data menu but I forget where.

Thanks. Excel isn't the problem I'm having, it's conceptualizing the relationship between price and production.  They move percentage-wise in magnitudes that are incompatible, so you have to dampen the price magnitude by several factors.

I've tried indexing them to a "100" number but then how do you choose what month corresponds to "100" for both except for arbitrarily?

Maybe I'll post some of my best aborted attempts later so you can see what I mean.

Maybe there isn't anything relevent we can draw from the two? Maybe there is, I just can't get a graph to show it.

That's why you want two axis - it lets you have both variables on difference scales on the same graph.
Yeah, you're right, I'm gonna try that next.