By September 1944 there were eight independent Armoured/Tank Brigades in 21st Army Group, shared between the five Corps (I, VIII, XII, XXX & II Canadian Corps). These brigades were 4th Armoured, 8th Armoured, 31st Tank, 33rd Armoured, 34th Tank, 6th Guards Tank, 2nd Canadian Armoured & 1st Czechoslovak Armoured Brigades. There had also been the 27th Armoured Brigade (disbanded in Normandy), while 34th Tank Brigade was in the process of being transferred to 79th Armoured Division to become an all-Crocodile brigade.
The Tank Brigades listed above were equipped with Churchill Infantry Tanks, while the Armoured Brigades were equipped with Shermans, except for the Czechs who had Cromwells.
The doctrinal line between 'Tank' and 'Armoured' Brigades was blurred to the point of non-existence by 1944, simply because there weren't enough Infantry Tanks (i.e. Churchills) to go around. There were only enough Churchills for three brigades (plus 1st Assault Assault Bde RE) in NW Europe, two in Italy and one (in the process of being created) in India. The remaining 20-odd brigades were equipped with medium tanks and were designated as 'Armoured' though still had the same infantry-support role (the three brigades in Burma being the exception - designated as 'Tank', but being equipped with a mix of Sherman, Lee/Grant and Stuart, though replacement with Churchill was underway for one of the brigades).
XXX Corps in Market-Garden had the independent 8th Armoured Brigade under command (4/7th Dragoon Guards, 13/18th Hussars, Sherwood Rangers Yeomanry, 12th KRRC & 147th Fd Regt RA) and was further reinforced by 44th RTR from 33rd Armoured Brigade and 15/19th Hussars from 11th Armoured Division. All of these units were initially under the command of 43rd (Wessex) Division, but some of the armour was detached along the way to support the US Airborne Divisions (44 RTR & 15/19 H to 101st Airborne and SRY to 82nd Airborne). There were also some additional Recce and Armoured Car units attached from other formations. There was no 'Funny' support from 79th Armoured Division, as they were busy dealing with the fortified Channel Ports at that time.
In terms of MT, an infantry division had roughly enough 3-Tonner TCVs to lift one of its three brigades at a time (or one battalion in each brigade). This could later be supplemented by 'Kangaroos' from 79th Armoured Division (there were eventually five squadrons of Kangaroos, each of which could lift a battalion). Armoured Divisions had enough TCVs to lift its single infantry brigade and Armoured Brigades of Armoured Divisions (as well as a few independent brigades, such as the 8th Armoured Brigade) also included a 'Motor Battalion', which was fully mechanised with halftracks and carriers.
[edited, as I'd said seven brigades and then listed eight...]