How can this bash script be made shorter and better? Is there a way to combine multiple curl statements and if statements in bash?

Is there a way to combine multiple curl statements and if statements in bash?

curl can retrieve multiple URLs in one run, but then you're left to parse its output into per-URL pieces. Since you want to report on the response for each URL, it is probably to your advantage to run curl separately for each.

But you can make the script less repetitive by writing a shell function that performs one curl run and reports on the results:

test_url() {
  local status=$(curl -o /dev/null -s -w "%{http_code}\n" -H "Authorization: token abc123" "$1")

  if [ "$status" = 200 ]; then
    echo "$2 is running successfully"
  else
    echo "Error at $2. Status code: $status"
  fi
}

Then running it for a given URL is a one-liner:

test_url https://url_1.com url1_1
test_url https://url_2.com url1_2

Altogether, that's also about the same length as the original, but this is the break-even point on length. Each specific URL to test requires only one line, as opposed to six in your version. Also, if you want to change any of the details of the lookup or status reporting then you can do it in one place for all.

To avoid repetition, you can encapsulate code you need to reuse in a function.

httpresult () {
    curl -o /dev/null -s -w "%{http_code}\n" -H "Authorization: token abc123" "$@"
}

check200 () {
    local status=$(httpresult "$1")
    if [ "$status" = "200" ]; then
        echo "$0: ${2-$1} is running successfully" >&2
    else
        echo "$0: error at ${2-$1}. Status code: $status" >&2
    fi
}

check200 "https://url_1.com/" "url_1"
check200 "https://url_2.com/" "url_2"

Splitting httpresult to a separate function isn't really necessary, but perhaps useful both as a demonstration of a more modular design, and as something you might reuse in other scripts too.

I changed the formatting of the status message to include the name of the script in the message, and to print diagnostics to standard error instead of standard output, in accordance with common best practices.

The check200 function accepts a URL and optionally a human-readable label to use in the diagnostic messages; if you omit it, they will simply contain the URL, too. It wasn't clear from your question whether the labels are important and useful.

Notice that the standard comparison operator in [ ... ] is =, not == (though Bash will accept both).