Word for more than several, but less than many

A 'dozen' is exactly 12, but you can use that for approximation, as well: 'there are around a dozen' or 'approximately a dozen'.

Alternatively, you can just say 'around 10', 'around 15', etc.

You can also say 'over 10' or 'at least 10' which implies that the number is close to 10.

The phrase quite a few may work. Curiously, it suggests more than a few but less than many. However, it is very vague and context dependent.

All these terms are vague; there is no precise number to them, so there is no accurate comparison.

However measure words sometimes have somewhat predictable comparisons. 'several' is definitely more than 'a few', and 'many' is less than 'most'.

But between 'several' and 'many'? Those are fairly synonymous, with several only working with smaller integers and 'many' applying to pretty much any scale (it is more relative). 'some' is relative and definitely less than 'many' but

I have some marbles

How many do people have? I don't know maybe a hundred at most, so 'some' might mean here 10, 20, maybe 30?

I have several marbles

This means I have more than just a handful maybe even 10, or 20, or even 30.

On a different scale consider

There are some Muslims in India

India has a population of a billion, this would lead you to believe that there might be a subset of that (maybe millions?).

There are several Muslims in India

this makes it sounds like there are under a hundred, quite a different thing than millions.

That should tell you that there is no real, exact answer to your title question.

To the implicit question in the contents, you're saying that both 'several' and 'many' are too much. Then use 'some' or 'a few' or nothing at all. Even 'a number' works (no one would so pedantically think of 0, 1 or 2 as a number when you say that, it is mostly synonymous with 'some'.

TL;DR

So I suggest:

There are a number of studies that...

probably to be held as truthful should be three or more. Ibky two would be disingenuous.

There are studies that...

This doesn't quantify much at all, it is informally used as 'not one, but I want you to think more, but probably only two.