URL as query param in proxy, how to navigate?

Question

I'm using http-proxy to create a simple proxy: localhost:3000?q=${encodeURIComponent(targetURL)} will access targetURL using the proxy.

Here's my working code:

var http = require("http");
var httpProxy = require("http-proxy");

//create a server object:
http
  .createServer(function (req, res) {
    try {
      // Get the `?q=` query param.
      const url = req.query.q;
      const parsed = new URL(decodeURIComponents(url));

      const proxy = httpProxy.createProxyServer();

      // Fix URLs and query params forwarding.
      // https://stackoverflow.com/questions/55285448/node-http-proxy-how-to-pass-new-query-parameters-to-initial-request
      proxy.on("proxyReq", function (proxyReq) {
        proxyReq.path = parsed.toString();
      });

      proxy.on("end", () => res.end());

      // Avoid the following error:
      // "Error [ERR_TLS_CERT_ALTNAME_INVALID]: Hostname/IP does not match certificate's altnames"
      req.headers["host"] = parsed.host || undefined;

      proxy.web(req, res, { target: url }, (err) => {
        throw err;
      });
    } catch (err) {
      res.write(JSON.stringify({ error: err.message }));
      res.end();
    }
  })
  .listen(8080); //the server object listens on port 8080

When I visit localhost:3000?q=https://google.com, everything works. However, if I click on a link in the website, then the route is changed on my hostname directly, not in the query param.

So:

• I go to localhost:3000?q=https://google.com
• I click on "Images', which should bring me to localhost:3000?q=https://google.com/images
• instead, it brings me to localhost:3000/images?q=https://google.com, which 404s

How do I solve navigation in the target website?