I have applied for a UK visitor visa (standard) for myself, my wife and 2 children which got rejected. I want to ask if can I submit my application first, and if it got accepted then submit my family's applications to save visa fees? Because if my application gets rejected they all will be rejected again.
Yes, you can. Whether you should depends on the reason(s) for the refusal and whether you can address it in your next application. Even if your next application is approved, it does not guarantee that your subsequent family applications will be, in particular if having family ties at home while you are in the UK is a factor in the approval. If you post a copy of the refusal notice with personal details blacked out, you will get a better answer.