I've travelled by coach from Singapore up to Kuala Lumpur before when I was in my twenties. I felt safe then and would probably do it with my kids then, so that could be a relatively cheap trip from Singapore
In Singapore itself there's so much to do. My kids really enjoyed the art science exhibition near marina sands.
If history is your thing, there is the Changi prison museum. It tells the story of prisoners of war from WW2 and is well worth a visit if you have the time.
Sentosa Island is also great.
A curry buffet lunch or afternoon tea at the raffles hotel is pricey, but well worth a visit.
There's a Lego land just over the border in Malaysia, which has great public transport links from Singapore
If you wanted a beach break there's Langkawi and redang island in Malaysia (on opposite sides of the peninsula- so have different rainy seasons, so check this before picking which to go to) Or if you fancied something further afield you can easily fly to Bali or Lombok in Indonesia
As for turbulence, I always tell the kids to make sure they've been to the toilet and strapped back in to their seats by the time the flight has reached the opposite side of India, as soon as the plane enters the Bay of Bengal things can get a little rough. The incident the other day is worrying, but I will still enjoy the flights I have booked for August.
Hope your travel plans all work out.