When I was delivering car across the country, I sometimes had to get to a destination close to Bristol Airport, approaching from the M4. I left the M4 at Jn 19, taking the M32 to Jn 1 and then following the A4174 outer ring road. (That's 1st exit from Jn 1, follow the A4174 until it reaches the the A4, turn right onto it, then go left back onto the A4174 again. Follow the A4174, which involves taking a right turn at a roundabout - so watch the road signs - until reaching the A38, which takes you to the airport.
That might sound a bit complicated but it basically comes down to this:
Jn 19
Jn1
First exit
Follow A4174 signs until picking up signs for 'A38 (Airport'
Chris