WebSocket connection with a ESP-01S (ESP8266) and arduino

I'm new to working with WebSockets and arduino. I'm trying to connect to a WebSocket server that I run on a separate computer from my ESP-01s. So far I'm able to connect to my wifi and get a local IP. I also tested connection between my Arduino and the server using a GET request.

My problem is that when I try to implement a WebSocketClient connection It just doesn't work. I'm using an Arduino UNO and a ESP-01S.

Here's my code:

#include <ESP8266WiFi.h>
#include <WebSocketsClient.h>
#include <ESP8266HTTPClient.h>

const char* ssid = "my ssid";
const char* password = "my pass";
const char* webSocketServer = "my ws ip";
const int webSocketPort = 3000;
const char* targetIP = "my ws ip";  
const int targetPort = 3000;
 
WiFiClient client;
WebSocketsClient webSocket = WebSocketsClient();

void setup() {
  Serial.begin(57600);
  Serial.println("Initializing...");

  // Connect to WiFi
  WiFi.begin(ssid, password);
  while (WiFi.status() != WL_CONNECTED) {
    delay(1000);
    Serial.println("Connecting to WiFi...");
  }
  Serial.println("Connected to WiFi");

  // Connect to WebSocket server
  Serial.println("Attempting to connect to WebSocket server...");
  webSocket.begin("echo.websocket.org", 80, "/");

  webSocket.onEvent(webSocketEvent);
 
  makeGETRequest();
  Serial.println("Setup complete");
}

void loop() {

  webSocket.loop();
}

void webSocketEvent(WStype_t type, uint8_t * payload, size_t length) {
  switch(type) {
    case WStype_DISCONNECTED:
      Serial.println("Disconnected from WebSocket");
      break;
    case WStype_CONNECTED:
      Serial.println("Connected to WebSocket");
      break;
    case WStype_TEXT:
      Serial.printf("Received payload: %s\n", payload);
      break;
  }
}



void printIPAddress() {
  // Get and print the local IP address
  IPAddress ip = WiFi.localIP();
  Serial.print("IP Address: ");
  Serial.println(ip);
}

void makeGETRequest() {
  // Create a WiFiClient object to establish a connection
  WiFiClient client;

  // Create an HTTPClient object
  HTTPClient http;

  // Construct the complete URL with the target IP and port
  String url = "http://" + String(targetIP) + ":" + String(targetPort);

  Serial.print("Making GET request to: ");
  Serial.println(url);

  // Begin the HTTP request
  if (http.begin(client, url)) {
    // Send the GET request
    int httpCode = http.GET();

    // Check the HTTP response code
    if (httpCode > 0) {
      Serial.print("HTTP response code: ");
      Serial.println(httpCode);

      // Print the response payload
      String payload = http.getString();
      Serial.println("Response payload:");
      Serial.println(payload);
    } else {
      Serial.println("Failed to get HTTP response");
    }

    // Close the connection
    http.end();
  } else {
    Serial.println("Failed to begin HTTP connection");
  }
}

I've tried to connect to both my own WebSocket server (which I've sucessfuly connected to from other PCs) and public servers such as "echo.websocket.org" but I havent been able to establish a connection (I get the message "Disconnected from WebSocket").

I am expecting the message "Connected to WebSocket".

What am I doing wrong? I'd really appreciate any help.