This answer has correctly been given twice before in this thread, so perhaps a third time will cement it in the questioners minds.
Lets set it out for you.
Provided you do have motor vehicle insurance, and the insurance provider has been informed and agreed the vehicle will be used for towing (covering modifications to the motor vehicle such as fitting of a tow bar) Then the vehicle insurance cover will automatically provide third party cover for the trailer whist being towed.
Even if you have fully comprehensive cover for your car, it breaks down into two types of cover. The first and legally essential part is :-
The Road Traffic Act requires all motorised vehicles used on the road, must have insurance to cover against third party claims. Whenever a vehicle tows a trailer, the trailer becomes effectively part of the 'vehicle' as far as the road traffic act is concerned, and as trailers cannot be insured as 'vehicles' on their own, it falls on the motor insurers to cover vehicles and trailers in tow.
The second part of a fully Comp is the additional cover you purchase for all the bits of personal loss you might encounter, such as the value of the vehicle and personal belongings etc. This usually excludes any attached trailer and their contents, but that might depend on the insurance package you buy.
If you wish to insure the value and the content of a caravan, then generally you have to purchase a separate policy.
Read the Terms and Conditions of the policies you have to establish what is covered, or consult your insurance provider.