Upright row for Lats? No, there a deltoid and trap exercise. The bent over row, is for the upper and middle back with inner palms down (or knuckles out and wide grip), and another variation is, palms UP, and a little narrower grip, and doing these by pulling more to the waist. And, yes if you have just free weights to use, these are good back exercises.
Another variation, is using the DB bent over row. (palms inward or out), with one hand on a bench and bent over.
PB, take a LITE DB (real lite), and try a lite variation of the DB bent over row.
Even if there is NO WGT on the DB, and see how it feels. Any pain (inappropriate pain, stop immediately)
I suggested before, that you see a doc. You may have injured other things connected to the lat you are not aware of, and the doc may be able to prescribe something to speed the healing along. If you decide that you dont want to do this, then you have made the choice to go down the potential same path that has led you to the point where you are at young man. Remember, PB, you are only 16, you DO NOT want an unknown injury festering while your body is still in the stage of overall bodily growth......now change your mind, youngman.
PB we are products of our past history, bottom line. Now get with it! ROCK WITH THE DOC..........or you will be left back on the Dock waiting on the clock
If you get your mind correct on this, your body will respond accordingly in time............do what it takes to get it healed, see a doc, PB.