Most rockets launch with a thrust to weight slightly over 1, because the engines are the most expensive part of the rocket, it’s usually most cost effective to add fuel until you can barely lift off. As most of the rocket is fuel, more thrust = bigger rocket, and more thrust also tends to mean a bigger flame. It’s not quite that simple because different fuels result in more or less visible exhaust. Hydrogen for example is nearly invisible, while solid rocket motors are very visible. The space shuttle has both, so you can get a pretty dramatic comparison.
Ambient air pressure also matters, at higher altitude the exhaust expands and cools off much faster, making it less visible.
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