Before starting, you need to be clear about the target you are going to proxy for. It shall be an URL.

Add the following code anywhere of your project.

        public static HttpRequestMessage CreateProxyHttpRequest(this HttpContext context, Uri uri)
        {
            var request = context.Request;

            var requestMessage = new HttpRequestMessage();
            var requestMethod = request.Method;
            if (!HttpMethods.IsGet(requestMethod) &&
                !HttpMethods.IsHead(requestMethod) &&
                !HttpMethods.IsDelete(requestMethod) &&
                !HttpMethods.IsTrace(requestMethod))
            {
                var streamContent = new StreamContent(request.Body);
                requestMessage.Content = streamContent;
            }

            // Copy the request headers
            foreach (var header in request.Headers)
            {
                if (!requestMessage.Headers.TryAddWithoutValidation(header.Key, header.Value.ToArray()) && requestMessage.Content != null)
                {
                    requestMessage.Content?.Headers.TryAddWithoutValidation(header.Key, header.Value.ToArray());
                }
            }

            requestMessage.Headers.Host = uri.Authority;
            requestMessage.RequestUri = uri;
            requestMessage.Method = new HttpMethod(request.Method);

            return requestMessage;
        }

This method covert user sends HttpContext.Request to a reusable HttpRequestMessage. So you can send this message to the target server.

After your target server response, you need to copy the responded HttpResponseMessage to the HttpContext.Response so the user’s browser just gets it.

        public static async Task CopyProxyHttpResponse(this HttpContext context, HttpResponseMessage responseMessage)
        {
            if (responseMessage == null)
            {
                throw new ArgumentNullException(nameof(responseMessage));
            }

            var response = context.Response;

            response.StatusCode = (int)responseMessage.StatusCode;
            foreach (var header in responseMessage.Headers)
            {
                response.Headers[header.Key] = header.Value.ToArray();
            }

            foreach (var header in responseMessage.Content.Headers)
            {
                response.Headers[header.Key] = header.Value.ToArray();
            }

            // SendAsync removes chunking from the response. This removes the header so it doesn't expect a chunked response.
            response.Headers.Remove("transfer-encoding");

            using (var responseStream = await responseMessage.Content.ReadAsStreamAsync())
            {
                await responseStream.CopyToAsync(response.Body, _streamCopyBufferSize, context.RequestAborted);
            }
        }

And now the preparation is complete. Back to our controller:

        public async Task<IActionResult> Rewrite()
        {
            var request = HttpContext.CreateProxyHttpRequest(new Uri("https://www.google.com"));
            var response = await _client.SendAsync(request, HttpCompletionOption.ResponseHeadersRead, HttpContext.RequestAborted);
            await HttpContext.CopyProxyHttpResponse(response);
            return Ok();
        }

And try to access it. It will be proxied to google.com