Oh the joy of bulk proofreading. Half of making it go quickly is simply practice. The more you do it, the easier it is to spot things. But in terms of tips:
-Set some goals for your proofreading. What are you watching for? Spelling? Grammar? Word flow? Word or syllable count (like for poetry)? If proofreading is an assignment or something, then specifically watch for whatever is requested of you.
-List out those above things you're watching for, then make that the focus of your proofreading. A piece of writing may be written averagely overall, but if it's mechanically sound then that's not your concern. Your job as a proofreader (I assume) is to not fix or improve their creative vision, but to simply clean up what they've written and make it as presentable as possible.
-I'll add some emphasis: reading someone's piece for enjoyability and how they can improve that is different than proofreading, which usually only focuses on writing mechanics. A full review of something involves both sides of it. Something can be enjoyable but have a ton of mechanical issues. Or something can be really cleanly written, but bland to read.
-When proofreading short prose pieces, I'll commonly do it from bottom to top so I'm focusing more on the mechanics of each line, rather than what's actually being written as a whole. Poetry I always read top to bottom regardless, since there's usually more rules and word flow involved.
-Fix up the grammar so it's appropriate to your grade level. Even teachers make small errors, and one could spend all day trying to research every obscure rule to make something perfect. Even after all the research, certain rules can be unclear because there's inconstancies between different countries and how rules are taught. Try not to overthink it, otherwise you'll start spending too much time getting caught up on little things. If it reads cleanly to the best of your knowledge, then it's likely good enough for your classmates.
-I'll usually proofread something 2-3 times before calling it done. The first time I broadly look for major issues and fix them. The second time I look for minor issues and fix them. The third time is more optional, but I like to do it as a final look to make sure it looks acceptable. By layering the process in 2-3 sweeps, instead of doing it all at once, it helps me not get too caught on singular lines which can slow you down the more you reread the same one. This is more a personal preference. Other people may be perfectly fine proofreading top to bottom in a single lookover.
-Reading out loud can be helpful for some people in spotting issues.
-Sometimes you'll notice certain habitual mistakes that crop up more often than others. It can be easier to spot an issue if you anticipate it'll keep happening.
-Lastly I'll echo mizal here. If someone is clearly putting no effort into their writing, don't feel inclined to put much effort into your proofreading either. Put in the effort that they put in. Just catch some of the big/obvious things and that's good enough, as long as you're meeting the expectations of your role. Quick proofreading falls under 1 sweep, instead of doing the full 2-3. On the flip side, if a couple people are clearly working hard on their writing, then acknowledge that effort with extra time making it as clean as possible for them (as time allows).
Edit: To add a little more to that last point. It's mainly about managing your limited time in ways that's most helpful for as many people as possible. If you had unlimited time and energy to proofread everything, then I always encourage people give their 110% at it even if some of the writing is less than inspired. When under strict time constraints, then it's important to manage it wisely and prioritize extra effort where you know it'll be most productive for everybody you're helping out.
