I’m probably looking for a book but I’d have had to put it in CS and more engineers tend to lurk here. Anyway, I’m looking for a book or website, ideally for laymen, that illustrates the “typical” solutions to “typical” engineering issues. Regulating flow of water through a tube, switches for different situations, etc… Do such books exist? I know of the website “How stuff works” and the book " How Things Works" which are good, but are always specific to some device.
I’m not clear on what you’re looking for here. Do you want reference on the mentality and process of engineering, or are you looking for some kind of “General Engineering Cookbook”? For the former, check out books like Henry Petroski’s books, or the excellent A History of Mechanical Inventions.
If the latter, every field and subfield has its references. For mechanical design engineering, for instance, Mark’s Standard Handbook For Mechanical Engineers, Machinery’s Handbook, and Shigley and Mischke are the standard texts; the former being a comprehensive (but somewhat shallow) survey on most fields relating to mechanical engineering, the latter being more specific technical information about machine design and simple mechanisms. Even these texts, though, are insufficent for coverage of industry-specific areas of information; a guy who designs cars is going to have very different texts on his shelf than the bloke who is making plastic dohickeys, or the lass who is making satellite chassis.
Every general field of engineering has its own set of references; while the fundamental physics is the same, the base of empirical knowledge is unique to the specific field.
Stranger
The above books are all great and I have all of them in my collection. But the one I used that made people think I was a genius just three years out of school was "Standard Handbook of Engineering Calculations " by Tyler Hicks. It covers mechanical, electrical, chemical, aeronautical and civil engineering. It has loads of practical examples and relevent calculations that will help you impress the ladies and make your friends envious. 
I also have that book and it is fantastic.
I tried looking the book up on Amazon but there isn’t much info about it. The word “Calculations” makes me wonder if it isn’t a list of tables. Is that not the case?
At $200+ It must be PHANTASTIC! Look for the big river price, and/or ‘used’ copies as low as $50.
No, it’s a book of sample problems typically encountered in many engineering fields.
Also of interest may be Pocket Ref, which is a little book of lots of info you may need when designing stuff. The Mythbusters swear by it.
Me personally, I swear that the selective application of enough explosives will solve just about any problem.
Tripler
And beer will help with those it doesn’t.
I’ll third that one. Machinery’s Handbook is also nearly indispensable. I have two editions; the first one is a 1941 (printed in wartime London) edition from before screw threads were standardized into the forms they have today. Very interesting from a historical aspect.
I’d also recommend this guy, Illustrated Sourcebook of Mechanical Components by Robert Parmley.