The First Minister has said he has received "vile messages from people who are opposed to the 20mph policy".
Speaking in the Senedd during First Ministers questions he told the leader of the Welsh Conservatives, Andrew RT Davies, that he would be happy to share with him some of the correspondence he has received.
He added: "It think it would be good to hear from him (Mr Davies) that those people who are prepared to say things that are not simply disparaging but are directly threatening of people's physical safety - that those things are absolutely unacceptable as well."
Andrew RT Davies responded by acknowledging "there is no space in our society for that whatsoever."
He continued: "What I would say to you, and to all politicians in this chamber, is that this type of language is completely unacceptable, deplorable, and should be called out at every opportunity."
The heated discussion came as a petition against the Welsh Government's 20mph policy has gathered over 430,000 signatures.
It has picked up more signatures than any other in the history of the Senedd, and was raised by Andrew RT Davies during First Minister’s Questions.
He told members that the deputy Transport Minister, Lee Waters said on Friday night in a tweet that the '440-odd thousand signatures to that petition, were anti-road safety.' The Leader of the Conservatives then proceeded to ask the First Minister in the Senedd chamber, "Do you agree with him?”
Andrew RT Davies, the Welsh Conservative leader questioned the First Minister at Tuesday's (26 September) plenary.
Mark Drakeford said: “All petitions should be taken seriously and that is exactly how the petition, in relation to the 20 mile per hour zones will be viewed by this government.
“The process is the one set out by the Llywydd that petitions are to the Senedd, not to the Welsh Government, and it is for the Senedd through the petitions committee, to respond to them. But I can assure the member, that we on this side, take all petitions seriously.
"I will view these petitions in that light. At the heart of the case for moving to 20 mile an hour speed limits as a default in built-up residential areas is road safety.
“This is a measure that will save lives, that is the basis on which it has been brought forward. And that is the basis on which we will continue to defend a measure which was only introduced last week.
“The leading academic journal in this field said it was the most significant public health measure to have been attempted in the United Kingdom for nearly 20 years.”
Earlier the Senedd’s Llywydd, or Presiding Officer, confirmed that the petition "won’t get an early debate" in the chamber.
Any petition that gains more than 10,000 signatures is put forward for a debate on the floor of the Senedd chamber once it closes.
The current petition isn’t due to close until March 2024 and unless the organiser agrees to finish it early, it won’t be considered for debate until after that date.
Elin Jones told MSs: “In this case, there is currently no request for the petition to be closed at this stage, and this may be the case until the six-month window closes in March 2024.
“I merely wanted to make all Members aware of us following due process with this petition, as with all other petitions.”
Want a quick and expert briefing on the biggest news stories? Listen to our latest podcasts to find out What You Need To Know…