Aim for the largest file that still fits
The single biggest mistake is choosing a target far below the limit that applies to you. If the form accepts 2 MB, there is no benefit in forcing the picture to 300 KB, and every kilobyte you shave below what is needed is detail you gave away for nothing.
Read the requirement, set that number as your maximum, and stop there. The compressor works toward a file at or below the target and reports the final size, so you can confirm the result rather than assume it.
Reduce pixels before you squeeze bytes
A photo displayed at a modest size rarely needs camera-resolution pixels. Resize it to the dimensions it will actually occupy, keeping the aspect lock on, and the compressor has far less work to do at the same target.
This is the route to a small file that still looks clean. Rather than pushing the encoder harder and harder on a huge image, you remove data that was never going to be visible in the first place, which is a much gentler way to lose bytes.
Compress from the original, not from a copy
Every time a JPEG is saved, it is re-encoded and some detail is lost again. Compressing an already-compressed file stacks those losses, so start each attempt from your best original rather than from the file you downloaded last time.
The same applies to screenshots that have been edited and re-saved several times. Fewer passes between the source and the final file means cleaner edges and smoother gradients in the result.
Inspect the result at full size
Look at the preview the way the image will really be seen: open it full size and check the areas that fail first, such as text, faces, hard edges, and smooth skies or walls. Small thumbnails hide damage that is obvious when you zoom in.
Compare the original and output numbers on the result screen as well. If the size dropped far below your target, you went further than necessary, and a second pass with a higher maximum will treat the picture more kindly. If the result looks soft, raise the target and try again from the original file rather than accepting the loss.
Know what the output will always be
The compressor writes a JPEG, so the result is always a lossy file: the goal is no visible loss rather than no loss at all. A transparent PNG background becomes white on the way through, because JPEG cannot store transparency.
There is no batch mode, so each image is compressed and checked on its own. That takes a few extra clicks, and it gives you the chance to judge every file individually before it goes anywhere.