How many patients do you have?
One thing you can do is keep a large notecard on each patient. You can't keep all the details of daily progress on there, but can keep a name, hospital number, chief complaint, a 1-liner HPI and perhaps a list of the important meds and the most important imaging and other studies the patient has had while in the hospital. Don't try to write everything on there, though.
Every day you need to have a list of your patients (hope your hospital has EMR that will print this?) and before rounds in the a.m. jot down any new lab and test results for that a.m. Try to have as many daily notes as you can written before rounds (not always possible, but try) or at least a skeleton note. If your attending doesn't care, you might be able to finish/fill in some of these while the other intern is presenting on rounds, or while the attending is taking a phone call, etc. One mistake I made as an early intern was trying to write my notes in a detailed way, like I did when I was a med student...sometimes you just need to shorten them because sometimes nobody is really reading your note anyway. The important parts are the oneliner about what happened overnight, the vitals and exam, the labs/new test results, and a short problem list with assessment/plan which includes billable information
(i.e.
1) CHF, decompensated, but improving - continue diuresis with IV lasix 40mg IV bid,
2) HTN - now controlled with PO lisinopril and metoprolol,
3) hypokalemia - likely secondary to diuresis. Replete with 40mg IV KCl, and restart PO KCl 20mg/day (home dose)
4) dispo - consider home tomorrow if SOB and edema further improved. Will discuss w/attending).
There's not one perfect answer to this problem. Part of it depends on how the services work at your hospital. You can ask a friendly resident (either yours or another team's) for tips on how to do things efficiently in your hospital.
When you are presenting, it's the first sentence or two that the attending really listens to. So act confident and try to have a reasonable plan for every patient. If you're not sure what it should be, try to ask the resident the night before or early in the a.m. before rounds...but not 1 minute before rounds or something. It's hard.
Another thing that helps is to shrink your census by sending patients home when they are ready. This means you have to think about dispo/discharge planning from the day they come in. This one your resident needs to be helping you with also (and hopefully you have a decent hospital social worker, too).
Good luck. We've all been there, done that.