How to count the number of times specific characters are in a string

First, and to generalize the method, it will be better if calc_fit() takes the array of letters as an argument too. Then, you can create a Map from the array with the counter of each letter starting on 0. Finally, you traverse the string and increment the respective counter of each letter when needed.

function calc_fit(element, fitness_let)
{
    // Create a Map from the array of letters to search.
    let map = new Map(fitness_let.map(l => ([l, 0])));

    // Traverse the string and increment counter of letters.    
    for (const c of element)
    {
        if (map.has(c))
            map.set(c, map.get(c) + 1);
    }
    
    return map;
}

let res = calc_fit("This is a string with some letters", ["e","l","m","n","t"]);
res.forEach((counter, letter) => console.log(`${letter} => ${counter}`));
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You can count occurrences with an identical value of an array using map and filter:

let str="I love JavaScript and Node.js ";
let arr=str.replace(/[^a-zA-Z]/g, '').split('');

const mapped = [...new Set(arr)].map(a => `${a} occurs ${arr.filter(a1 => a1 === a).length  } time(s)`);
console.log(mapped);