Hey Chilli
There's a lot of things going here that could contribute to your Bhajis breaking up. This could be due to the mixture, the temperature of the oil, or the pan you are using.
I think the first thing I would need to know as are you doing flat Bhajis (e.g. like squashed fishcakes in appearance) that you are shallow frying or are you doing Pakoras (e.g. more like round balls in shape and deep fried).
But going back to basics I would say firstly your mixture should be able to hold it together in the precooked stage. That's you don't want a loose watery mixture. I aim for a not too thin batter that is a bit like yoghurt in consistency. The other thing is that the oil needs to be of a temperature where it will immediately cook the outside, thereby sealing the Bhajis or Pakoras and thereby holding it's shape. However if too hot you get instantly burned, bitter pieces and raw mixture on the inside, so I would say oil at 170/180 should be in the right ballpark.
These are just my initial thoughts but if you post your recipe here I might be able to add some more comments as will others. There's a wealth of experience on this site so I am sure you will get the help to move your Bhajis forward.