I am using the CM-Unicode version of the font Computer Modern Serif, and the left double quote looks bad.
CSS:
@import url("ComputerModern/serif/cmun-serif.css");
body {
font-family: "Computer Modern Serif", serif
}
HTML:
“conscience”
Result:
Is there any way that I could fix this, or is it just an error on the CM-Unicode people's end?
As answer to your question: you can open the font in any half-decent font editor, find the glyph for the opening double quote, fix the pathing, and then save and use the fixed font. Although, if you're going down that route then what you should actually be doing is of course to just file an issue with the people that maintain this font so that they can fix it for everyone, forever, not just for you, for now.
However, the solution to your problem is to not use Computer Modern. CM was designed for print, not for screen media (that wouldn't be an option for decades to come when CM was designed) and this port looks absolutely dire. It's super jagged and clearly no one bothered to put in the hours necessary to get the most out of the OpenType port. If you want to use a "modern" font and you like the look of Computer Modern, then give the OpenType version of Latin Modern a try instead.
Compare the weird Computer Modern font you're using to the proper Latin Modern OpenType font:
And of course remember that if you know the sizes you're using on your page, you'll want to make sure to pick the most appropriate font: default browser size is 16px, so using the optimized-for-8-points lmroman8 will look better than using lmroman10. Conversely, using lmroman10 for text that's sized to 20px will look better than lmroman8, etc.