Transmission popping out of gear

check the drain plug magnet for metal bits . If any the box has to come out 4.th gear is straight shaft connection and 5 gear connects from the cluster shaft up to the out put shaft . On either side is a rear main bearing and from your description if that bearing is failed then the 4th gear syncro will not align properly and the gear for 5th will be out of alignment so the syncro to work will not work

If the Linkage is not damaged, you are going to need to pull the Transmission out.

The part that went bad is called the SYNCRO's

best advice is dont go 40mph in 5th gear... if you are going up a hill at 40mph in 5th gear i have no doubt that it is jerking, your putting too much stress on the engine. shift up to 4th or 3rd you will be fine. 5th gear is an overdrive gear. 55+mph

syncro worn ,easy job to change thoug providing its a manual if its an auto then its a new gearbox so trade it in quick

Your shifting block in the transmission is not working properly. replace it.

Check the operation/adjustment of the 5th/Rev gear gate and associated selection levers.
Check 5/R gear lever linkages are operating correctly.

gear linkage worn joints.

no. linkage most likely you need to get adjusted or
one of it is worn out

The gear linkages are not correctly aligned.
Is the gearbox mounting in exactly the same place as when you removed it?
Even a 1/4 inch will offset the gears.
hope this helps

hi i have the fault code here and it says gear 4 incorrect ratio.wiring fault,transmission switch,shift solenoids or mechanical faults in the gear box.hope this helps ,best of luck.