I guess these are the linear halogen bulbs LadyA?
I used to get this with site lights (building sites) until I went over to LEDs years ago.
Usually down to a poor connection at either end of the bulb. They can be an absolute pain to seat properly. I shouldn't recommend this, but, in desperation, I used to switch it on and twiddle the bulb around until it worked.
It didn't always, the contact ends in the lamp are spring-loaded. The connections were always going wrong.
I guess your old lamp's connections have worn out, and the new one's are just poorly made.
If no luck with twiddling, then I'm afraid it's time to upgrade. I honestly can't think of anything else, I'm afraid.