Two questions here that I would greatly appreciate any input on 
1. Is it possible for a US citizen visiting the UK on a visitor's visa to legally marry a UK Citizen?
2. Is it possible for a US citizen visiting the UK on a visitor's visa to apply and receive a fiancee visa while still in the UK - without need for going back to America first?
It seems to me that the visitor's visa has the same limitations as a fiancee visa. But any input would be great thanks! 
For the first question: yes. You can marry in the CofE, for example. To marry in England and Wales at a Registry, you will need a Certificate of Approval (COA). Section 3 of the COA asks if your visa is 6 months or more and if not, they are going to need a really and *really* good reason. The reason will need to be legally compelling on constitutional grounds and not something like "we can't stand to be apart".
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Constitutional grounds means that you need to convince them that they did not contemplate your circumstances when they made the law (I&A TOC 2004). So in practical reality, the answer is no.
For the second question, the answer is no, they are not going to let somebody convert from a visitor to a fiance or spouse. There was too much abuse of that loophole (mostly by Americans) in the 90's and 00's and they changed the law to forbid it (I&A 2002). There's a provision to allow exceptions, but once again it can't be something like "We love each other too much to abide by the rules".
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Generally it's not worth fooling around with exceptionals because you can damage your credibility.
Edit: You don't necessarily need to go stateside. You can go to any country where you have domicile. Ok, for most that's the USA, but not necessarily for every US citizen.