Whatever you do, don't say "SOUTH-WARK"

I don't want to appear too much of a tourist when I visit LONDON later this year.
That's why I am honing my pronunciation of certain place names.
I already knew that KEIGHLY up in the north of the country is actually pronounced "KEETH-lee," and even many non-Anglophiles know LEICESTER is pronounced "LEHS-ster."
Those two examples, however, cannot prepare one to know how to pronounce LEIGH. Is it "LEETH?" Is it "LEH?"
It's neither. It's "LIE."
So why do the English pronounce TEIGNMOUTH as "tin-MUTH?"
I haven't got a clue.
Everyone knows READING is "RED-DING," like the town in California.
Did you know the Norfolk town of HAPPISBURGH is actually pronounced "HAZE-burr?"
Hmm! Is that a fact?
Truthfully, there appears to be no rhyme nor reason to the pronunciation of English place names. I suppose they can say the words any way they wish. They did invent the language, after all.
SOUTHWARK? Yeah. The area of London pictured is actually pronounced "SUH-thik."