The other answer about the Trump presidency is not entirely correct. While Trump employed Tillerson, there was a plan, however unrealistic, to still oust Assad by diplomatic means:
Obama’s old policy, such as it was, had suffered a catastrophic meltdown when confronted first with reality and then with Russia; and Trump had failed to string together even three coherent sentences on Syria during his presidential campaign.
For the first few months of the new administration, policy drifted in no particular strategic direction. By late summer 2017, then-secretary of state Rex Tillerson had finally come up with a plan that promised to take the president’s varied interests into account while in fact recommitting to the anti-Assad line and rejecting a U.S. pullout. Trump signed off on it that autumn, momentarily swayed or simply distracted.
The public rollout came in a January 2018 speech at Stanford University, in which Tillerson vowed to stay engaged in Syria and oust Assad by means of non-military pressure. The secretary explained that the United States would keep its troops embedded among the Kurds of northeastern Syria to prevent a jihadi resurgence, and would also ramp up economic pressure until Damascus accepted a “post-Assad future” that would neither attract jihadis nor permit “malicious Iranian influence.” [...]
But it wasn’t the mismatched means and ends that felled the Tillerson plan. It was simmering conflict between the secretary and his boss (whom he had, famously and allegedly, called a “moron”), in addition to Trump’s instinctive dislike for mucking around in the Middle East. Almost immediately after Tillerson’s Stanford speech, Trump began to question why the United States still had troops in Syria, apparently feeling he’d been conned into accepting something he didn’t like and that he didn’t think U.S. taxpayers should be paying for.
Tillerson was sacked in March 2018. Around the same time, the president reportedly read a Washington Post story about Tillerson’s promise of limited economic aid to Kurdish-ruled areas outside Assad’s control, which sent him into a fit of rage. Out of nowhere, he froze the U.S. stabilization budget for Syria, saying the Gulf oil kingdoms could pay for it if they wanted it so much, and demanded that U.S. troops should come home “as soon as possible.”
And eventually that is going to happen, modulo a detour through Iraq.
But back to the story line, Tillerson's departure was not a complete abandonment of any strategy in Syria, but rather it became mainly an anti-Iranian strategy.
Over the summer of 2018, U.S. officialdom started to assemble the president’s latest preferences into a new plan to replace the one ejected alongside Tillerson. But like some indestructible squishy toy, Syria policy immediately began to reassume the open-ended and interventionist form that Trump had just denounced.
One reason was that the president’s hard-ball tactics had worked—sort of. Arab and European leaders winced at the prospect of a quick withdrawal, and when U.S. diplomats came panhandling, allied nations agreed to fund the stabilization of former jihadi-held areas in order to let American taxpayers off the hook without derailing the Syria mission. The president was pleased.
Another reason was the surging influence of citizens concerned by the Iranian land bridge. As he moved to exit the “decaying and rotten” Iranian nuclear deal in spring 2018, Trump started to stuff his government with Middle East hawks who saw Syria primarily through an anti-Iran lens. That April, Trump replaced his national security adviser H. R. McMaster with John Bolton, an advocate of bombing Iran and overthrowing its regime, while another top-tier Iran hawk, CIA chief Michael Pompeo, was sworn in to succeed Tillerson.
In office, Bolton and Pompeo have renounced regime change in Tehran, instead settling for a policy of “maximum pressure” that is supposed to pave the way for direct talks with Iran’s leaders and a new, better nuclear deal. The concept seems patterned on Trump’s North Korea fliplomacy, and so may have originated in the Oval Office.
However, Bolton’s real views are well known—he doesn’t want a better deal, he wants regime change—and Pompeo’s list of preconditions has been so expansive—Iranian withdrawal from Syria being just the appetizer—that they amount to asking the ayatollahs to come out with their hands over their heads. Rather than engaging with American demands, the Iranians have therefore spent summer and autumn showing how they can inflict asymmetric damage, threaten U.S. interests and allies, and disrupt oil markets. As for evacuating Syria, forget about it: “We stay there as long as Syria wants [it] so,” said a commander of the Islamic Revolutionary Guard Corps, the paramilitary force that controls Iran’s Syria deployment. [...]
By early September , Trump had re-approved plans for an indefinite stay in Syria, with even more fanciful goals than those previously formulated by Tillerson. “We’re not going to leave as long as Iranian troops are outside Iranian borders, and that includes Iranian proxies and militias,” Bolton explained.
Perhaps it shouldn't come as a big surprise that Trump's latest decision to withdraw from Syria, which is being implemented, came after Bolton was fired in September 2019.
“It was in Bolton’s nature to run an imperial NSC [national security council] but he stepped on the toes of too many people,” said Mark Groombridge, who worked for Bolton for a decade. “He got into the crosshairs of Pompeo and Mulvaney, who saw Bolton as a liability for the 2020 election. War on every front was not what Trump ran on.”
And I almost forgot about Mattis' resignation and that Bolton did dig the Kurds, angering Erdogan:
President Trump first made waves on the issue in December 2018, when he abruptly announced the U.S. would completely pull its troops from Syria, tweeting that the mission to defeat ISIS was completed. The move prompted the resignation of then-Defense Secretary Jim Mattis and a coordinated campaign by then-National Security Adviser John Bolton to try to protect the Kurds, America's fighting partner in the region.
Trump’s foreign policy decision on Syria, which he had promised back on the campaign trail in 2016, was reportedly the final straw for Mattis. His decision to resign was based on principle, sources told Fox News at the time.
Bolton and other officials in the White House worked “behind the scenes” to slow the president’s order to pull all 2,000 troops from Syria, with Bolton pushing for the U.S. departure from Syria to include the condition that Turkey guaranteed it would not target Kurdish fighters.
Turkish President Recep Tayyip Erdogan accused Bolton of making a “serious mistake” in complicating Trump’s Syria withdrawal — and Bolton would ultimately also exit his post after being fired by Trump in September.