The president also specifically called out the El Paso shooting suspect, who on Saturday morning opened fire at a Walmart and killed 20 people. Thirteen hours later, another gunman killed nine people in downtown Dayton, Ohio.
Trump also asked the Federal Bureau of Investigation to identify what resources the agency needs to investigate and disrupt "hate crimes and domestic terrorism", and said mental health laws must be reformed.
Trump, who has been accused of not doing enough to tackle domestic extremist groups, said he would direct the Department of Justice to investigate domestic terrorism and would propose legislation to ensure that those who commit hate crimes and mass murders face the death penalty.
The author says he's held these beliefs for years, before Trump became President.
Less than a week after the mosque shootings, New Zealand moved to ban "military-style" semi-automatic weapons and high-capacity magazines; similarly, after a mass shooting in 1996, Australia enacted sweeping gun bans within two weeks. It defines such an event as a shooting in which at least four people are killed or wounded.
Democratic congresswoman Rashida Tlaib has managed to spark intense debates online as she moved to criticise the reaction of US President Donald Trump's advisor and daughter, Ivanka, to the mass shootings in Texas and OH that rocked the United States over the weekend.
"We must recognise the internet has provided a unsafe avenue to radicalise disturbed minds and perform demented acts", he added.
"The Internet, likewise, is used for human-trafficking, illegal drug distribution and so many other heinous crimes", Trump continued.
The president said he's enlisting the help of social media companies to aid in early detection.
Responding to Trump on Monday, Senate Democratic leader Chuck Schumer said if the president was serious about strengthening background checks, he should demand Senate Majority Leader Mitch McConnell "put the bipartisan, House-passed universal background checks bill up for a vote".
Many people took exception to Trump's mention of "many sides" after a protester ran down a group of counter-protesters with a auto, killing one.
DeWine has been endorsed by the National Rifle Association (NRA) for his pro-gun stance.
Authorities identified the El Paso suspect as 21-year-old Patrick Crusius from Allen, a Dallas suburb which is a almost 10-hour drive from El Paso.
Trump says America must stop the glorification of violence in society.
Trump delivered the speech-his first public appearance since the tragedies-Monday morning from the White House flanked by Vice President Mike Pence.
Trump has denied responsibility for inciting violence in the past. When referring to the suspect's manifesto, he added that it was "consumed by racist hate".
Trump's name or rhetoric has been cited in other cases relating to violent acts, threats of violence or allegations of assault.
Trump did not directly address accusations that his anti-immigrant and racially charged comments have contributed to a rise in hate crimes. I mean, to link - these shootings were clearly in part result of his racist rhetoric. "He's not tolerating racism - he's promoting racism".
Apple CEO Tim Cook expressed outrage over the shootings on Sunday, saying it's time to "come together to address this violence for the good of our country".
"Hate has no place in our country", he said.
We've seen, very recently, how determined most Republicans are to avoid criticizing the president.