The answer is Surprisingly THIN. If I recall correctly from a column or post written by Bernard Levine, 19th century 'Bowie' and other large knives of the frontier were almost never more than 3/16 inch thick, and most were thinner than 1/8 inch.
i was very surprised to learn this,though
i forgot where i saw it,but IIRC i saw something on it on the histroy channel. they were talking about different tool/goods that fur traders used.
i expected them to be 3/8 or 1/4 inch thick, boy was i wrong.(not the first time
Not so quick. Then, as now, there has always been a wide variation of knives lumped under the Bowie designation. The butcher knives were a little thinner maybe 1/8 - 3/16", but actual survival / sidearm / sporting bowies were heavier and sometimes much heavier. I have seen plenty of examples with blade thicknesses of about 1/4", and some have been spoted with 5/16 or heavier blades.
You might be confusing the thickness of the guard, which was usually thin with the blade thickness. The big thick 1/4" plus guards, with heavy lugs etc., are a 20th century invention.