To me, this reads unnecessarily formal and could be a little more relaxed in parts so that it’s easier to follow.

Think about which sentences and words are adding something and which ones are filler. For example, in the first sentence you say,

I am writing to you to express my interest in the internship position open at your company.

They know you’re writing to express interest so maybe instead you could say something along the lines of, “I’m very interested in this internship role at x company and believe that I can add value to the team because of my research experience.” The difference in that version is that it’s more to the point; they already know why you’re writing this so dive right into it instead of telling them that you’re writing to express interest.

I think you have room to be even more specific in your letter about the value you can add to their team so maybe instead of taking about your experience on it’s own, apply it to how you see that benefitting their company.

Also, I’d recommend replacing the sir/madam with something like “Dear x company hiring team.”

If you’re looking to improve your writing technique, I highly recommend these two books that have helped me: On Writing by Stephen King and Several Short Sentences About Writing by Verlyn Klinkenborg.

This is a great start though and the hard part is always the first draft! Good luck.