String replace Double quotes into curly brackets

Try using a global regex and use capture groups:

let str = '"This" is my "new" key "string"';
str = str.replace(/"([^"]*)"/g, '{$1}');
console.log(str);

The "([^"]*)" regex captures a ", followed by 0 or more things that aren't another ", and a closing ". The replacement uses $1 as a reference for the things that were wrapped in quotes.

Your code currently is only working for the first occurrence of each { and }. The easiest way to fix this would be to loop while there is still a " in str:

let str = '"This" is my "new" key "string"';
while (str.includes('"')) {
  str = str.replace(/"/,'{').replace(/"/,'}');
}
console.log(str);