It's a data volume problem, not necessarily an architecture problem.
Whether on 1 machine or 1000 machines, if you end up summarizing 1,000,000 rows, you're going to have problems.
Rather than normalizing you data, you need to de-normalize it.
You mention in a comment that your data base is "perfect for your purpose", when, obviously, it's not. It's too slow.
So, something has to give. Your perfect model isn't working, as you need to process too much data in too short of a time. Sounds like you need some higher level data sets than your raw data. Perhaps a data warehousing solution. Who knows, not enough information to really say.
But there are a lot of things you can do to satisfy a specific subset of queries with a good response time, while still allowing ad hoc queries that respond in "10-20 minutes".
Edit regarding comment:
I am not familiar with "GridSQL", or what it does.
If you send several, identical SQL queries to individual "shard" databases, each containing a subset, then the simple selection query will scale to the network (i.e. you will eventually become network bound to the controller), as this is a truly, parallel, stateless process.
The problem becomes, as you mentioned, the secondary processing, notably sorting and aggregates, as this can only be done on the final, "raw" result set.
That means that your controller ends up, inevitably, becoming your bottleneck and, in the end, regardless of how "scaled out" you are, you still have to contend with a data volume issue. If you send your query out to 1000 node and inevitably have to summarize or sort the 1000 row result set from each node, resulting in 1M rows, you still have a long result time and large data processing demand on a single machine.
I don't know what database you are using, and I don't know the specifics about individual databases, but you can see how if you actually partition your data across several disk spindles, and have a decent, modern, multi-core processor, the database implementation itself can handle much of this scaling in terms of parallel disk spindle requests for you. Which implementations actually DO do this, I can't say. I'm just suggesting that it's possible for them to (and some may well do this).
But, my general point, is if you are running, specifically, aggregates, then you are likely processing too much data if you're hitting the raw sources each time. If you analyze your queries, you may well be able to "pre-summarize" your data at various levels of granularity to help avoid the data saturation problem.
For example, if you are storing individual web hits, but are more interested in activity based on each hour of the day (rather than the subsecond data you may be logging), summarizing to the hour of the day alone can reduce your data demand dramatically.
So, scaling out can certainly help, but it may well not be the only solution to the problem, rather it would be a component. Data warehousing is designed to address these kinds of problems, but does not work well with "ad hoc" queries. Rather you need to have a reasonable idea of what kinds of queries you want to support and design it accordingly.