Header BrowserRouter içinde olmadığı için olabilir bir de alttaki gibi denermisin.

import HeaderComp from "./components/HeaderComp";
import { Switch, Route, BrowserRouter } from 'react-router-dom';
import MainPage from "./components/MainPage";
import HelpPage from "./components/HelpPage";
import DownloadPage from "./components/DownloadPage";
import SearchPage from "./components/SearchPage";
import DonatePage from "./components/DonatePage";

function App() {
  return (
      <BrowserRouter>
        <div>
          <HeaderComp />
          <main>
            <div>
                <Switch>
                  <Route path="/" exact>
                    <MainPage />
                  </Route>
                  <Route path="/download" exact>
                    <DownloadPage />
                  </Route>
                  <Route path="/search" exact>
                    <SearchPage />
                  </Route>
                  <Route path="/help" exact>
                    <HelpPage />
                  </Route>
                  <Route path="/donate" exact>
                    <DonatePage />
                  </Route>
                </Switch>
            </div>
    
    
          </main>
        </div>
    </BrowserRouter>
  );
}

export default App;