Just curious, I am not an American citizen, I am from Belgium.
Just because your “President” Trump signs a bill, is it automatically a law?
Does he really have that kind of power? Using Twitter maybe?
Frightening, no?
The classic cartoon linked above, well-known here in the USA, does a pretty good job of explaining the process, but avoids all the backroom deals, corruption and bribes that sometimes occur to get laws passed. We have far too many laws, many of them just flat-out purchased by one special interest or another to get what they want.
To give you a simple answer though, no - the President's signature on a bill is the final step of a long process, after a bill (called a "bill" until it's finally a law) is passed by both our House and Senate (the two branches of our Federal Congress). Then it goes to the President for signature, but he has the option to NOT sign it, called a veto, stopping it from becoming a law. A veto can go back to the Congress and be overridden into law without the President's agreement, but it requires far more supporting votes to do it. This system, called Checks & Balances, is supposed to prevent any single part of our government from unilaterally having its way without broad support from the others.
Our President has NO authority whatsoever to create or change laws on his own; he can suggest laws and changes to laws to Congress, but they have to do it - the President's primary responsibility is to manage the implementation and enforcement of laws.
Since I was very young and first grasped how our system is set up via a method created in 1791 (when our Constitution was adopted), I've been thoroughly impressed by, and proud of, the insights and wisdom of those men long ago - they did a pretty darn good job creating a system from scratch. And they wrote down their thinking and reasoning along the way, so we know what was in their minds as it unfolded over several years. Many disagreements, many arguments, but they finally hammered out an agreement for the ages.
These days, as with this silly registration, it's getting harder to remember that our USA government was created to secure the individual liberties of free, self-governing citizens. Millions of laws that now exist hide that noble goal.
End of sermon, but any questions feel free to post or private-message me. Thanks for your interest in our process.