Let's say I have an URL like this:

/city/nyc   (display info about new york city)

and another like this:

/city/nyc/streets   (display a list of Street of nyc)

I can bind them to a method like this:

Route::get('city/{city}', 'CityController@showCity');
Route::get('city/{city}/streets', 'CityController@showCityStreet');

The problem is that I need to execute some checks on the city (for example if {city} is present in the database) on both methods. I could create a method and call them in both like this:

class CityController {

    private function cityCommonCheck($city) {
       // check
    }

    public function showCity($city) {
      $this->cityCommonCheck($city);

      // other logic
    }

    public function showCityStreet($city) {
      $this->cityCommonCheck($city);

      // other logic
    }
}

Is there any better way?

I think best way to do this, you can move common logic into a Model.So your code would like below.

class CityController {

  public function showCity($city) {
      City::cityCommonCheck($city);  
  }

  public function showCityStreet($city) {
    City::cityCommonCheck($city);
  }
}

model class

class City{
    public static function cityCommonCheck($city) {
      //put here your logic
    }
}

In this way you could invoke cityCommonCheck function from any controller.

Even though you think differently, I believe a middleware is the best solution for this.

First, use php artisan make:middleware CityCheckMiddleware to create a class in App/Http/Middleware. Then edit the method to do what your check is supposed to do and add a constructor to inject the Router

public function __construct(\Illuminate\Http\Routing\Router $router){
    $this->route = $router;
}

public function handle($request, Closure $next)
{
    $city = $this->route->input('city');

    // do checking

    return $next($request);
}

Define a shorthand key in App/Http/Kernel.php:

protected $routeMiddleware = [
    'auth' => 'App\Http\Middleware\Authenticate',
    // ...
    'city_checker' => 'App\Http\Middleware\CityCheckerMiddleware',
];

Then, in your controller:

public function __construct()
{
    $this->middleware('city_checker', ['only' => ['showCity', 'showCityStreet']]);
}