A few questions to help get a better idea of your situation ...
Where in your mouth is the bridge and how many units are involved?
What was in this area before this bridge was made? How long ago were the teeth extracted?
Were the adjacent teeth sensitive when you wore the temporary bridge or did they become sensitive after the permanent bridge was cemented on?
Does the bridge have a metal substructure?
Is the sensitivity more to hot or cold temps ... or equally to both?
Does anything hurt or feel uncomfortable when you bite down or chew food?
What is the general health of your other teeth, gums and jaw bone?
Teeth sensitivity that begins after dental work can occur for a variety of reasons. I will give you a list of some of them:
1) The teeth being crowned were previously restored and/or unhealthy to begin with and cutting them down for the crowns resulted in further inflammation to the nerves inside the teeth.
2) The preparation of the teeth could have been too severe resulting in excess removal of healthy tooth structure leaving the teeth vulnerable or susceptible to irreversible pulpitis.
3) The bridge may be ill fitting around the gingival margins of the crowns leaving gaps large enough for the teeth to react to different temperatures.
4) The bridge may not fit properly within your existing bite which would cause inflammation in the nerves and the periodontal ligaments of those crowned teeth resulting in sensitivity or pain.
The reason the root canals are recommended is to stop you from complaining about the problem. I know that sounds harsh, but it is the typical recommendation when the dentist does not want to explore what the real problems are OR when the dentist suspects that the inflammation in those teeth is irreversible. The problem with doing the root canals is that the procedure itself does not prevent future problems from occurring with those teeth and is really only a temporary measure to quiet the current symptoms.
Root canal therapy severs the blood vessel that supplies the tooth life and nutrients rendering the tooth non-vital. It also somewhat removes only the visible nerves from the inside of the large visible canals. All teeth have many many hundreds of microscopic canals, called dentin tubules or accessory canals, that not only contain nerve tissue but are inaccessible. Therefore, all root canaled teeth are essentially without life, without nutrition and contain dead nerve tissue within the dentin tubules. I will attach a diagram showing the dentin tubules so you can get a visual of just how much nerve tissue remains inside of each root canaled tooth. The likelihood of an abscess occurring after root canal therapy is highly likely at some point. Especially if the root canaled teeth are anchor teeth to a bridge due to the pressure they endure as anchor teeth.
Unfortunately, you have limited options. First it needs to be established WHY the teeth have become problematic. If the problems are irreversible, then the only options you have are to extract them or do root canals. Extracting them removes the source of the problem but you may need to replace them with dental implants or a removable partial denture. Root canaling them is at best a temporary measure to quite down the current symptoms.
If the teeth are healthy and just severely inflamed from all the dental work ... it may be possible to remove the bridge, put temporary crowns on the anchor teeth and figure out a different plan to replace the missing teeth. The anchor teeth would need to be monitored for a few months to be sure there is no infection brewing and then if all looks well they can be covered with well fitting permanent crowns. But it's important to know that any future dental work on these teeth is risky because the nerves are always going to be vulnerable to further inflammation.
You have a very common dental situation but it's not an easy one to remedy.
Your dentist should have informed you, if he didn't, of the possible problems prior to moving forward with the bridgework.
***I have been in the dental profession for over 3 decades. I am an educator, trainer and extensively experienced in chair side assisting and dental radiography. The information that I provide here is my opinion based on my professional experience and is not meant to be taken as medical advice.***