They don't stop replication. It's just that you replicate your dna in a number of cycles, let's say 30. after which you'll have a approximatly 10^6 molecules of the amplified region and only a few longer strands.

If you draw it out you'll see that your region of interest quickly start to dominate.

original strand
----/  /----ppp-------------------------/  /--
----/  /-----------------------qqq------/  /--

after one cycle
----/  /----ppp-------------------------/  /--  
            -------------------qqq------/  /--

----/  /----ppp-------------------
----/  /-----------------------qqq------/  /--  

after two cycles
----/  /----ppp-------------------------/  /--  
            -------------------qqq------/  /--

            ppp-------------------
            -------------------qqq------/  /--

----/  /----ppp-------------------
----/  /-----------------------qqq------/  /--  

----/  /----ppp-------------------
            -------------------qqq

after three cycles
----/  /----ppp-------------------------/  /--  
            -------------------qqq------/  /--

            ppp-------------------             
            -------------------qqq------/  /--

            ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq------/  /--

----/  /----ppp-------------------
            -------------------qqq 

----/  /----ppp-------------------
----/  /-----------------------qqq------/  /-- 

----/  /----ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq

after three cycles
----/  /----ppp-------------------------/  /--  
            -------------------qqq------/  /--

            ppp------------------- 
            -------------------qqq------/  /--


            ppp-------------------             
            -------------------qqq

            ppp-------------------             
            -------------------qqq------/  /--

            ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq------/  /--

----/  /----ppp-------------------
            -------------------qqq 

----/  /----ppp-------------------
            -------------------qqq 

            ppp-------------------
            -------------------qqq

----/  /----ppp-------------------
            -------------------qqq------/  /-- 

----/  /----ppp-------------------
----/  /-----------------------qqq------/  /-- 

----/  /----ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq

            ppp-------------------
            -------------------qqq

etc.