Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Turk seems to have heavy constant panting and I understand that you are concerned for him.
Panting can indicate difficulty exchanging oxygen, getting overheated, nausea, pain, anxiety or excitement.
Please check his gum and tongue color. They should be a nice bubblegum pink.
If you press firmly on his gums, and then quickly let go the gum should blanche (whiten) and then return to a nice pink color in less than 2 seconds.
If his gums blanche and then return to pink, and are bubblegum pink then her oxygen levels are likely fine and he is relatively stable.
If they are gray or white then possible heart disease, internal bleeding, red cell destruction or lack of production would be possible and he should be checked by a veterinarian promptly.
Dogs with allergies usually have itchy skin and ears, they rarely have respiratory signs. They can breathe with difficulty secondary to respiratory tract irritants like cigarette smoke or scented oils of candles, but I would think that there is more going on with him than that given his breed and the usual conformation issue they suffer from.
If you are hearing a lot of noise that is is likely due to mucous mixed with air produced when he struggles to breathe through abnormal airways.
Bulldogs and other flat faced breeds are known for having a complex of issues that can make breathing more difficulty, called brachycephalic syndrome. These issues include an overlong soft palate, stenotic (small, closed in) nares, and parts of the larynx that evert out when breathing called everted laryngeal saccules.
If he has a low grade respiratory infection or is exposed to cigarette smoke, pollution or other irritants then that can be a possible complicating factor as well.
Turk needs to be evaluated by your veterinarian. They can determine how much he is affected and whether surgery to correct some, or all, of these problems would be beneficial for him.
Dogs that don't have these problems corrected surgically can suffer from exercise intolerance and easily suffer from heatstroke because they cannot breathe properly. Long term they can suffer from tracheal collapse and secondary heart disease from their struggle to exchange oxygen. If the abnormalities are corrected surgically the pup can have a normal life.
Surgery is best done by a very experienced, board certified surgeon to have optimal results.
Here is a link to a webpage that discusses brachycephalic syndrome if you'd like more information: https://www.acvs.org/small-animal/brachycephalic-syndrome
In the meantime I recommend making sure Turk does not get overheated or exercise too strenuously. If he is starting to open mouth breathe then play should stop.
Best of luck with Turk, please let me know if you have any further questions.