Yes, if you follow the ILS GP down, you will be a bit below the PAPI, progressively getting lower and lower, because the PAPI beams fan out from one point. So what you will see is exactly what you did, 2W2R into 1W3R, then 4R very close in.
As I said up-thread, the PAPI is further into the runway to cater for long-bodies (as per FD's diagrams). The 738 does not need it, and people criticising others for being off the PAPI when they are spot on the GS are off the mark. At the very least, your trainer should have explained what you have read here so you understand the issue.
And as Flying Stone mentions, using the PAPI for active flight path control below the ILS DA isn't really practical anyway.
Our AIP has this:
Note the difference between 3W and 3R; the equivalent of 2W2R to 4R. That's why you were low on the PAPI, because that's where the GS is.