Welcome to health care magic!
I have gone through your question and from my experience I'd suggest that you have a rather deep cut on your palm.
Had I been at your place I'd surely see a doctor once atlest and get my wound properly checked, as there are many small tendons in Palm which may be hurt due to this cut.
Moreover, you should have a shot of ATS ( anti
tetanus serum), if you didn't have one within last six months for any other reason.
Till you see a doctor, I think it would help you if you clean the wound nicely under running water and put "salon" ( as
savlon doesn't burn!) or any other antiseptic to prevent infection there.
Turmeric is considered to be a natural antiseptic but one has to be sure that it is a good brand or home made and not the poor quality having colors or other chemicals which might harm!
As it is a deep cut both in length and depth, may be you'll need a stitch to keep it in place and heal properly but one can be sure only after looking at it.
If I had been your doctor, I would give a pain killer like
Voveran, may be 50mg for it and an antibiotic cover.
hope this helps you taking care of your
injury.
feel free to ask if you have any other question regarding this.
Dr. Manisha Gopal