Timing belt went out and messed up engine valves

At highway speeds it may have bend some valves, is this a 4cyl engine or a v-6? this all depends on the mileage and the condition of the car, the cylinder heads would need to be removed and valves replaced plus a new belt and waterpump and timing belt pulleys, this is costly, if its a 4cyl it maybe worth tryying a belt and see what happens, that dont have to put it all back together they can run the engine with a new belt and if you got lucky then put it back together.but if its the v6 engine i would not try just junk it, sorry or replace engine if car is worth it.

4 or 6 cylinder? you say "heads" so I assume its a V6? A interference engine does NOT require a new engine. unless upon disassembly there is severe piston damage. A 2003 Galant isn't going to be worth putting in a $ 2500-3000 new engine. A car that has had scheduled service usually doesn't break the belt, its replaced long before that happens. Yes, you can take the heads off and get it serviced, new valves etc. On a V6 you also want to replace the water pump.

The 2.4L four has a belt and is not an interference engine according to the Gates timing belt application chart for your vehicle (check it out on Gates.com.) The 2.7L V-6 doesn't show that it needs a belt--so it has either a timing chain and gears or just gears. If the belt broke on the 2.4L four, you should be able to put on a new belt and go. The engine should not be damaged.

If the timing chain on the V-6 broke then engine damage may have occurred. Chain engines are sometimes not listed as interference because breaking a chain is so rare. The damage would probably be bent valves and damaged piston tops. The labor involved in repairing all that might cost more than just getting a rebuilt engine and putting it in.

If the timing belt broke while you were driving then unfortunately yes the valves can get messed up. The piston(s) can hit the valve(s) as it comes to the top of it's stroke. Common for valves to get bent and less common for pistons to crack.

If you installed a new timing belt, I would make sure the valve timing is set and then do a compression test.

no remove the head then remove the camshaft ,turn the head upside down and fill the combustion chambers with oil and make sure the valves are sealing ,i do fell you only have bent valves

No you could be lucky depends on RPM at time of brakage but chances are the valves are bent a good mechanic can pull the head have it reworked and reinstall it for about $600 or you can buy a chilton
manual disconnect the exaust the intake pull the head have it reworked for about $225 $45.00 new belt $60.00 for a new head gasket cost $330.00 but you have to do it your self. But You May have been lucky a long time ago a friend of mine had a timing belt break on a fiat shop told him $300 this was in 1987 I bought the car from him for $50.00 dollars $15 dollars more than the junkyard would give him I bought a timing belt for $25.00 spent 3hrs installing it, Drove the car for a year and a half till one of the rear strut mounts tore loose still a good deal but I was lucky had valves been bent i would have lost 75 dollars in87 that was 1/2 a weeks pay wish you the best of luck oh by the way most engines bend valves
when the timing belt / chain break

You have damaged the Valves so the head will have to be removed and recondtioned at a machine shop or buy a rebuilt at auto supply house, the head will cost $250.. with an exchange head. you will have to replace pushrods for the valves if so equipped, and antifreeze oil and filter lifters as well so figure about $500.00 to $800.00 to do the fix to make it run right, they will punt in a new water pump and a new timing belt tensioner and belt and other things that you will not understand.

I do not have good news for you.
According to the Gates Timing Belt guide 2002 Edition, both engines offered in the Dodge Neon are interference engines.
This means that your engine has likely sustained sgnificant valve, cylinder head or even piston damage. Have you tested the compression?
It is probably least expensive to buy a used engine, change its timing belt and replace your engine with that one.

If the timing belt has broken it's more than likely you have some valve damage.
Not the end of the world.
Options are...Head overhaul, inc valve replacement.£££££££££
Or find a complete Cylinder Head in a breakers yard and swap.

cheapest option complete head strip, should be able to order new valves, just hope you havent cracked or put a hole in a piston, easiest (and most reliable) option complete recon head (not cheap)but is guaranteed to be perfect. id say as for only 40 000, if you have reciepts, and it wasnt too long ago it was done you may have warantie comback and itll all be done for free.