No, this should work, I do it all the time.
If you have a really old iPhone it may close other apps if you use another one, killing Strava. Which iPhone do you have? When you switch back to Strava does it look like it's restarting the app (e.g. showing the loading screen for a long time) or just switching back?
It could also be a permissions issue - check that Strava is allowed background location access in the Privacy settings.
(although I note on my phone Strava is set to "Only when using the app" and it works fine in the background, but shows a blue bar at the top of the screen when recording in the background)