For an interviewee, an interruption can be very damaging. It breaks into your train of thought, throws you off course and reminds you all too forcibly that the interviewer is in charge and is taking control.
But it need not be like that.
First of all, why do some interviewers interrupt so much? If the interviewer is facing a politician, the interruption is probably intended to drag the interviewee back from his party message. It may be that the interviewer wants to unsettle a particularly bombastic interviewee. Or perhaps the reply has gone on too long and simply needs to be halted. Much more likely and much more mundane is the probability that the interviewer is running out of time and wants to ask a couple more questions.
All that should be of no concern to you, the interviewee, who simply have a point to put across. If you find yourself interrupted in the middle of a brief, relevant reply, you should insist on finishing. Do it politely but firmly – “If I could just finish my reply, because this is an important question….”. The interviewer is quite likely to back off but more importantly, the audience will be on your side. You are the person the audience wants to hear from, not Mr Clever Presenter.
So be firm, be polite and simply remind the interviewer that he is not in sole charge of this particular piece of broadcasting territory.